The integral of a rate of change of a quantity over 3 time interval ives the tOlal change in the quantity over the time in rva l.. • Solution to initial value problem.. In effect, the "a
Trang 1DEFINITIONS INTERPRETATIONS A '(x )= f(x ) (valid for one-sided derivatives
• !leuristies The definite integral captures the idea of
adding the values ora runclion over a continuulll
• Riemann sum A suitably weighted sum of values A
definie illlcgral is the limiting value of sueh sums A
Riemann sum of a function f defincd on [ II ,h is
determincd by a partition, which is a finite division of
[ {I h· 1 into subintervals, typically exprcssed by
{I =x u< x,"' < x,, =h ; and a sampling of points, one point
from each subintcrval, say c; frolll [x; " x;] The associated
"
Riemann sum is: ~f(ci)(Xi - X; d
j I
A regular partition has su intervals all the samc Icngth,
{\ =( h ~{I)/II , x;= {I + i {\'x A partition's norm is its
maximum subintcrval length A left sum takes the left
endpoint Cj=Xj_ of each ,ubinterval; a rght sum, thc
ri ht endpoint An upper sum of a continuous f takes a
p int <"; in each subintcrval where thc maximum \alue of f
is achieved; a lower sum, thc minimum value E.g., tl e
u per Riemann sum of cosx on [ 0 , 3 ] with a regular
partition of II intcrvals is the left sum (since the cosine is
decreasin on the interval: f I CO s( U- I :t )+ I ] :i.
3
• Definite integral The definite integral of f fol11 {I to b
may bc described as l 'j (x ) d x= lim ~f ( C i)Llsj
II 11lxll () i The limit is said to cxist ifsome number S (o be called the
integral) satisfies the rollowing: Every £ 0 admits a 6
such that all Riemann sums on partitions of [ tI ,h with
norm less than 6 dilTer from S by less than £ If there is
such a valuc S the I'unction is s id to be integrable and the
value is denoted l 'j(x ) dx or I f Thc function must be
bounded to be integrable The function f is called the
integrand and the points {I and h arc called the lo,ver limit
and upper limit of integralion, respectively The word
j
integral rerers to thc rormation of a f from f and [(I.h],
\Ve il as to the resuiting value iflhere is one
• ·\ntiderhative A "ntidcrivatvc of a fi.lI1ction f is ,[
functi n A whose dervative is f : A'(x ) =f(x) for all x in
some domain (usua y an interval) Any t'.I'O antiderivatives
of a functi n on an interval differ by a co stant (a
consequence of the Mean Value T henrem) E.g., oth
2 x - a - a ( i X " - (IX are anti cnvatlvcs 0 X - I ,
( lilcrlng y 2a- The indefinite integral ora runction!
den ted jj (x)dx , ' ,s an expression for the family or
allliderivatives on a typical (often unspecified)
I intena!' E.g., (o r x ~ I or for x > I)
j ' - - X - d x =!x - 1 + C
!x' -i
The constan C, which may have any real valuc, is the
constant of integration (Computer programs, and this
chart may omit the constant, it being understood by the
knowledgeahle user that the given anlidcrivative is just one
representative of a ramily.)
• Area under a cun·e I ('lis nonnegative and continuo s on
[a, h ] , then ( >t fed dx !;ivcs the area hetween the x-axis
•
and the graph The area function A (x) = .c ' f (n dl gives
the arca acculllulated up to x Iffis negative, the integral
is the negative of the area
• Average value.The awragc valli 011over an interval [a, h
i ' 1
Illay bc defined by avera ge VII[lI e ' h -=- / f(xldx
a (/
A rough estimatc of an integral may be made by estimatin the average value (by inspecting the graph) and multiplying it by the len th of the illlerva!' (See M II Ii' /l/I! Theore m ( MVT) jiJl" illtegra /s in the Theon ' sccti n.)
• Accumulated Change The integral of a rate of change of
a quantity over 3 time interval ives the tOlal change in the quantity over the time in rva l E.g., if v(l) =s '(I ) is a velocity (the rate ofehange of positio , then v (l )1'lt is th approximat displacement occurring in the time cremen / to t+{\./ ; adding the splacements for all time incrcmcn.ts
time interva l In the limit of small time incrcments, o e
."
gcts the exact total displacement: / v(f)dl=s(h)~-'(IIl
"
-Integral curve Imagin that a runc on f determines a slope fix) lor e ch x Placin linc segments with slo e
fi x) at p ints ( x.y) for variousy, and doing this lor various
x one gets a slope field An antidcri ative offis a function whose graph is tangent to the slo e field at each point The graph of the antiderivativc is called an integral curve of the slope field
• Solution to initial value problem The solution to the diiTerential equmiony'",/"(x) with initial valuey(xnl=y" is
( x )= y ,, + (' fll ) d l
x
THEORY
• Integrability & inequalities A continuo s functi n on clos d interval is integrable Integrability On [tI h implies integrability on closed su intervals of [tI h] Assumingfis
integrable, if L "'f(x) 5 M for all x in [{I ,hJ then
L · ( b -a lS l " f (xld X M (/)- a)
"
Use this to check integral evaluations with rough
o erestimates or u derestimates
M l~ ~ ~ ~~ ~ ~~ ~B " "~I
~ -L
I> ( ) Iff is nonnegative, then (".f x dx is n nnegative
Iff is integrablc on l/.hJ, then so isf ; and
i [ ' j(x) d x i< t if (x) l dx
• Fundamental theorem of calculus One part of the theorem is used to evaluate integrals: Iffis continuous on
[(I, h ] and A is an antidcrivative offnn that interval, then
j·" f (x) dx=A(x ) " C= (b) - A(a)
The other part is used to construct antiderivativ
If f is continuous on [a , h) then the functi
A (x)l= ( j U) dl is an antiderivative of f on [(I ,h]:
a
endpoints)
Fundamental Theorem
fl.x)
- - - - -::; - - ;.; ~ -- ;.o - -.
A(x)
• Differentiation of integrals r unctions arc oftcn defned a" integrals E.g., the "sine integral function" is
Si(x)= /,:"( S i l ) d
To dirferentiatc such, use the 'ccond part of t fundamental theorem: Si '(x)= sin x / \' A fu such as ( ' f (t)( l/ is a cOI11[losition in\olving
• a A(u )= ( " f ( tldt To difrcrentiate, usc the chain rule
a
J1 j" j W d l = J1 A (x ')= A ' (x')2x = 2.xj(x'1 '
·Mean value theorem for integrals Ir f and ~ arc continuous on [ II , h ], then there is a S in [a h such Lhat ,
In thc case g - I, the a\eragc value of f is attained
somewhere on the inLerval: -b- a a f (x) dx =.f( ; ) I
i\1VT for Integrah
~
• Cbange of variable formula An integrand and limit, of integration can be changed to make an integral easier to apprehend or cvaluate In effect, the "area" is smoothly rcdistributed without changing the integral;' value !I' g is
a function ith continuous derivative andfi, continuous then 1"f(" lI ) dn = I ' df (.9(f ))g' (f ) dl, where c d are points with g (C)= il and g(rI )= h
In practice, su stit!!!r lI g(l); compute t11/=g'(I)dl; and find what t is when 1/= and I/= h E.g I/ =sin t effects the
/ .", ,; I' ""~
transrormation of \ -u- d u= of ! - s in-I co~ I til
Oil 2
whkh becomes I cos ' l dl , since !1 -sin' l = c o s I
."
for ()< 1 < 1[/ 2 The formula is ollen used in revcrse, staJ1ing with ]," 1-' (.9(x ))O' Cd d x ec Technic/iII! s on pg
• i'atural logarithm A rigorous definition is In x =
(' Ida The change of variable formula with 11= I I
, U
yields / , a rill = , I-cc-1- d l = - _ I dt sho\\ ing that In( Ix) = ~I nx The other elementary propertie, of the natural log can likewise be easily derived from thi, definition in this approach, an inverse function is ded ced
and is derined to be the natural exponential function
,
n
Trang 2INTEGRATION FORMULAS
• Basic indefinite integrals Each lormula gives just one
antidcrivativc (all others dillering by a constant from that
given) and is valid on any open interval where the
integrand is defined:
~ 1/ ! 1
Jx"dx=~(nn+l * -1) f 1dx=lnlxl x
fe"xdx=e;x(k * O)
feos x dx=sin x fsin x dx= -cns x
• Further indefinite integrals The above conventions hold:
ftan x dx=lnlsec xl J~()t x dx=ln l sin xl
Isec x dx=lnlsec x + tan x l
fcosh x dx=sinh x fSinh X dx=cosh x
f ~=lln x'-a' 2a l x-al x+a Iix l dx= lxlxl
f l ~-+\: ,=lnlx+lx'+a' l =sinh 1~+lna
f ld~ ,=Inlx+lx'a'icosh 1~+lna
x-a
(take positive values for cosh· l )
flx' ± a'dx=!xl x'a' ± ~Inlx+ Ix' ± a' i
(Take same sign + or - throughout)
• Common definite integrals:
1 x"dx= - - j 'Ir'-x'dx= IT"- l "sin xdx=2
" n+l II 4 IJ
j 'rr " cos'8d8=1"n 1+cos20 d8=
" sin' (1d8=1" ; '1-cos28 d8=
TECHNIQUES
• Substitution Refers to the Change of variable formula
(see the Thmrv section), but ollen the formula is used in
f."F(y(x»)y' (x)dx (with F and g'continuous), you can put
II=g (x), tllI=g'(x)dx, and modify the limits of integration
In effect, the integral is over a path on the II-axis traced out
start], then the integral is zero.) E.g., u= I+x' yiclds
l l- _ x _ 1 -1 [I_ l_ ? d -11 (1+ ')
Substitution may be used lor indefinite integrals
E ' .g., f ~ l+x2 dx-lfdll-llna-lln(l+x')' -2 u - 2 -2 '
Some general formulas are:
f y(x) 1I (x)dx ,z:tl, y(x)dx-ln1y(x)l,
• Integration by parts Explicitly,
['u(x)u' (X)dX=Il(X)U(x)l~ - ['U(X)Il' Cddx
The procedure is used in derivations where the iemetions
arc gencral, as well as in explicit integrations You don't
with one factor to be integrated and the other to be
differentiated; the integral is the integrated factor times the
Other routine integration-by-parts integrands arc arcsinx,
Inx xnInx xsinx, XCOSX, and xe UX ,
• Rational functions Every rational function may be written as a polynomial plus a proper rational function (degrec of numerator less than degree of denominator) A
.A.! +",+~"
x-c (x-c)"
Al +Blx + + A"+B,,x x'+bx+c " (x'+hx+(-)'"
Math software can handle the work, but the following case
a-1
In gencral, the indefinite integral ofa proper rational function
fU - Idll,ju-"du(tI > O,fll(U'+ll "du (handled with
substitution II =tan t)
IMPROPER INTEGRALS
• Unbounded limits IIIis defined on [11,00] and integrable
provided the limit exists
E.g l w xdx= Iim(1-e 1J)=1
u /J iC J
In each case, if the limit exists, the improper integral
(_ 00,00) and integrable on every bounded interval,
f .~jlx)dx = ,[i~ .4 j(x)dx+ ,li~., j(x)dx (the
choice of c being arbitrary), provided each integral on the
• Singular integrands Iff is defined on (lI ,b] but not at
"" (hI Ih
), j(x)dx = }i!~ , j(x)dx provided the limit exists A
II ,,4-x' " - ' 1J ,,4-x'
lim arcsin(f)=!f,
Singular Integrand
t
E.g., fl ~dx= lim 1" ~dx+ lim [I ~dx if the
IX' a - ' II - IX' b ' 0 ·1, X'
diverges
• Examples & bounds
I x
2
-u1 x" x converg~s lor p < ,( Iverges 01 lcrWlse
· - ( ' I ~ )/dx convcn.!.cs for p> I divcr!.!cs otherwise
N t f d X -oe: x(lnx)" - - (1I - 1l0nx)" I I
I'P> cOJ1\crges at x=oo, p=O or < 1 div~rgL's at
'11
E.g 1 -;j1 dx converges to I and 1~dx dl\crg~s
x-The above integrals arc llseful in comparisons
establish convergence (or divergence) and to get
'1
E.g , 1 l': x' - dx convergcs since th~ integrand is
2:~ ,l ),dx = ) , + 2 < 2.4
APPLICATIONS
GEOMETRY
• Areas of plane regions, Consider a plane region admitting
."
[a,b] Sometimcs it i simpler to I' iew a region as bounded
integration variabk isy
~<X(P)-J~
a
• Volumes of solids Consider a solid
perpendicular to the
having square horiLollwi Cfllss-scctions with bottom ~ide
V = f~'( l - ~rdz = S~lh ,
·Solids of re\olution Consider a solid of r~I'()lution
sections (shells) or the solid parallel to ihc a is or
Trang 3I
• \rea of a surface of re".lution The surface generated by
revolving a graph y=I(x) between x=a and x=h about the
x-axis has area 1"2rr.f(x)/I+f'(xl'dx If the
I,
/ generating curve C is parametri7cd by «x(1) , 1'(1»
1c2rr.yds= " 2rr.yU) x' (t)-+ ý (t)-dl
PHYSICS
• '\Iotion in one dimen.ion Suppose a variable
displacement x(1) along a line has velocity v(t)=x'(t) and
acceleration ăl)=v'(t) Since v is 3n antidcrivative of a,
of an object thrown at time 1,,=0 líom a height x(O) =x"
with a vertical velocity 11(O)=VO undergoes the acceleration
-g due to gravitỵ Thus v(l) = Í(VII) + l' ( u)du = v,,-I:t
II
• \\orL If F(x) is a variable force acting along a line
parametrized by x the approximate work done over a sIIlali
displacement ~x at x is ~W=F(x)~" (force times
displacement), and the work done over an interval [ạh] is
In a nuid lifting problem, often ~W=~F'''(Y), where
cross-sectional arca Ăy) and width ~ỵ and the slab's
weight is ~F=pĂy)~y , p being the fluid's weight-densitỵ
-0
it m Solution to initial value problem; an example of that
type is in Motion in one dimension In those, the expression
for the derivative involved only the independent variablẹ A
basic DE involving the dcpendent variable is ý=kỵ A
gcneral DE where only the first-order derivative appears
and is linear in the dcpendent variable is ý+p(l)y =q(t)
Generally more difTicult arc equations in which the
independent variable appears in a Ihlt{ nonlinear} way;
DEs that are linear in the depcndent variabJe; c.g
y"=-kỵ x2y"+xý+x2y=Ọ
·Solutions A solution of a DE on an interval is a function
that is dillcrentiable to the order of the DE and satisfies the
equation on the interval It is a general solution if it
describes virtually all solutions if not all A general
solution to an 11th order DE generally involves II constants
each admitting a range of real values An initial value
problem (IVP) for anlllh order DE includes 11 specification
of the solution's valuc and II-I (krivativcs at some point
Generally in applications, an IVP has a unique solution on
some interval containing the initial value point
• Basic first-ord~r linl'ar DẸ Thc equation ý=ky,
rewntten lit =ky suggests y =kdt where l yFkt+c In
this way, one finds a solution y=CeAt On any open
interval every solution must have that form, because
interval Thusy=Cekl (C real) is the general solution The
unique solution with y(a)=y" is y=y"ek(l "J The trivial
solution isy Ọ solving any IVP y(a)=Ọ
• (.eneral first-order line r DE Consider
homogeneous equation h'+p(t)1r =0 (dhlh =-p(t)dt)
If .Í is a solution to the original DẸ then (ylh)'=qlh
where y=h fq / h The solution with y(<<)=y" is
'Tãlor pol~nomials The nth degree Taylor polynomial of
1 , f U ' I (c )( x - c)" (provided the dervatives exist) When
n.·
c=O, it's also called a MacLaurin polynomial
'Tãlor ' s Íormula Assume I has 11+1 continuous
interval Then lor any x in the interval.f(x)= P,,(x) +R ,, (x)
where R,,(x) = ( n ~l!!.1"''' I I(q) (X - C),,+I for somc S
between c and x (s varying with x) The expression for
the remainders for the Maclaurin polynomials ofI(x) =
(-1)"
In(l+x), -1<x<l, arc R,,(x) = I • X"+Ị
(n + 1) ( 1+;)"
There is a S between 0 and x such that In( I + x) =
• ror bounds As x approaches c the remainder generally
becomes smallcr, and a given Taylor p lynomial provides a
bettcr approximation of the function valuẹ With the
assumptons and notation above, if !f''' II (x)1 is bOlmded
"(fl~l!! Ix-cl" i I tor all x in the interval Ẹg for ~ " I <
<,x =l+x+x2 / 2, with error no more than ftlxl" = 0.5Ix!",
• Big 0 notation The statement f ( x)=p(x)+O ( x m )
f(x) - p i x)
(as x O) means that x '" IS bounded near x=ọ
(Some authors require that the limit of this rato as x
approaches 0 ex is!.) That is, f(x)-p ( x) approa hes 0 at
essentially the same rate as x'" Ẹg Taylor's formula implies I ( x )= f ( O) o f'(0)x+!f"(0)X 2+O( X 3 ) if I has
continuous third derivative on an open interval containing
from the identities in the item B asic M ac La u ri n Serie s ]
'Íllupital's rulẹ This resolves indeterminate ratios or
(H o r ~ ) IfFĨf(x)= 0 = F~g(x) and if IJt.nJ(x) = 0
= IJ'~: g(x) are defined and glx)"Ọ ror x ncar a (but no
the latter limit exists, or is infiniẹ The rule also holds when the limits ofIand g are infinitẹ Note that/'(a) and
g'(ll ) are not required to exist To resolve an indeterminate dill"erence (00_ 00) tr to rewrite it as an indcterminate ratio and apply l' Hlipial's rulẹ To resolve an indeterminate exponential (O".loọoroo"), take its logarithm
to get a product and rewrite this as a suitable indeterminate
ratio: apply L: II"pital 's rule; the exponential of the result
resolves the original indeterminate exponential
where lim I x l x = ell = Ị
x -0
NUMERICAL INTEGRATION
• General notes Solutions to applied problems often
involve definite integrals that cannot be evaluated easilỵ if
at all by finding antiderivativcs Readily available
software using refined algorithms can evaluate many integrals to lleeded precision The following methods for
approximating l"f(x)dx are elementarỵ Throughout II is the number of intervals in the underlying regular partition
·Trapezoid rulẹ The line connecting two points on the graph of a positive function together with the underlying illterval on the x axis l'lrl11 a trapezoid whose area is the
average of the two function values times the length of the interval Ađing these areas up over a regular partition
gives the trapezoid rule approximation
3
average of th left sum and right sum for th g 'en partition
The approximation remains valid iffis not posiivẹ
• \lidpoint rul~ This evaluates the Riemann sum on a regular partition with the sampling given by the midpoints
of e ch interval: 1 11" = " I ( a + 1 -2" h h Each
summand is the area of a trapc/oid \vh o ~l.· top is th tangent line segment through the midpoint
1'1,+ 211 (In the
• Slmp,on s rule The weighted sum:{ , :1 '
interval [a , h ] yields Simpson's rule
s = b -;;a (f ( a )+4 f(a ! h) + f(b»)
This is also the integral of the Simpson's ~;?
function at the three points For A
a regular parition of [ a h int
an even number /1=2", of ~-' -::-:;:""L
'-i!f1'
S2m= ::r U (a) +4 ;~/ ( a )+[ 2i + 1 ]" +2 ;~/(a+ 2í/I )+ fib)}
where " = (h - a) l lỊ impson's rul is e\a t 011 cubics
SEQUENCES
consist of all integers greater than or equal to somc initial integer, usually 0 or Ị The integer in a sequelll"e at /I i< usually denoted with a subscripted symbol like a" (rather
than with a functional notation ă/I» and is ealkd a term
of the sequencẹ A sequence is olien referred to with an
expression for its terms ẹg., 1//1 (with the domain
{l / n}, ;"" ,orn l , l / n(n = l, 2, )
) I Ĩnt SCllu, An arithmetic sequence un has a
u,,=a,, _ I+d=Uơd·/Ị It is a scqucllIial \(,fsion of n linear
l[ll1ction the common diflerence in the mil' of slopẹ A
geometric sequence, with terms Nfl" has a common ratio r
hetwcen successive values: gn=I:",lr=g"r" ẹg 5.0 2.5 1.25, 0.625 0.3125 It is a sequential "'r,ion of an exponential function the common ratio in the rok of basẹ
·ConH~rl!: 'nee A sequcnce (un] Co"\'erges ifsuml.! number
L (called the limit) satisfies the j(,lIolVing: l::.Iery £>0 admits all N such that la,,-LI< £ Í"·ullll" v Ifa limit L
exists, there is only one: on(' says i u,,: t :onn~rgc~ to L and
writes a,,-L, or ,!i!" all = L If a sequence does not converge, it diverges If a seqllclll'L' a" di"'~r g cs in 'lh::h a
way that every M>O admits all N suth that a,, > Jl h,r all
r= I then rll-+I; otherwise r" diverges and ifr> 1 r" - x ,
• Boun 'd n unotone Sl ph Il ;\11 iilcr~a:-.ing sequencé that is bounded above converges (to OJ limit less thall or
equal to any hound) This is a fundamental "let about the real numhcrs, and is basĩ to series convcrgt.:llcc tL·~t S
SERIES OF REAL NUMBERS
." 'r A ~crics is a scqucn :c ohtained by ađing the
\
values of another sequence L;a" = aợ +a, The Hduc
" u
ofthe serics at N is the sum ofvalucs lip to a, ·and is ~3lkd
a partial sum: La" "n+ +a The scries itself is
II II denoted La" The an arC called the terms of the series
II II -(00\ , t' A series L:a" converges if the sc:tJucnce or
" II partial slims converges in which case the limit of the
sequence of partial S UI11S is called the sum "Í the <eric,
If the series converges the notation Í.)r the ~eric s it elf stands also for its sum: La" = lim L\ a"
1/ II \ "
Trang 4Series continued
An equation such as La/I == S means the series converges
fI 4)
and its SlIlll isS In general statemcnts, La" may stand for
~
/I II
A form L;a,." where r is areal number and a" O A key identity
1/ I)
\ 1-.S ,
- is L;,," = I+r+r2 + +r;\= - _ I (, * 1) It implies
.01I1III L;,." =_I_(fl,.l ll(alsoL;al"" =a(_L - 1)) and
that the scries diverges if Irl > I The serics diverges if
r =±1The l'~Hlvcr ge ncc and rossiblc Sum l)fany geometric
series can b e determined lI ing thl.' pl"cccding k1J"l11ll1a
E.~ L; :f,, = 4 ! , <,- 1)= 2
II I 1 1
• p-scn', For P, a r~a l numbr L: II) is clled the I)-series
If I II
The /I-series diverges if ps i and converges if p>1 (hy
b low ) The harmonic series L; I~ diverges li,r the partial
" ,
" 1 IV
SUIllS are unbollnded: L: Ii ~ 1 + ~
1/ 1
• \lternafin '\eri~ , Thcse are series \\!hosc terms alternate
strictly decrease in ahsolutc valuc and approach a limit of
z ro, then the series (:onvcrgcs Moreover the truncation
error is less than the absolute value of the first omitted
term : IL:(- IY1 a ll - ±(- 1)fl a" II ' < a\ fI , (assuming
(11/ 0 ill a strictly Llecreasing manner)
CONVERGENCE TESTS
• Basic consideration" For any It: if L:a" converges, then
" A
Latl converges and conversely If a" 7 H, then L:all
" ,
diverges (Equivakntly irL:u lI convl'rgcs, then (In O) This
says nothing about e.g., L; 1 , ;\ series of positive terms is
" h II
partial sums i ~ hounded the ;;e l i e ~ COI1\crges This is the
foundation of all the following criteria for convergence
• Integral te,t & e~timat< Assume I is continuous,
p sitive nd decreasing on (K.oo) Then L;/(II) converges
" A
if and only if 1 /Ix)dx converges If the series
\
converges, then L; /(II )S L; f(n) + J /(x)dx the
right side o v cr~stimating the slim with error less than
J _ "1 1 _
L;" - L; , , + l " dx -1.2011l " the len side
,,\11 II III -'I., X
underestimating the sum with error less thanfl N+ I)
Int eg r a l test
,
"
"
"
" f( · \' +I)
K 'V '\'+1
~ ' 1 12 1 X> 1
E g " -;-\ - " -;-\ + 1 -;- \ dx 1.2018 " an
Z underestimate with error < 13.1<5.10 4
iU •Absolute comcrgence Ir L;ia"i converges, that is, if
Lall tcol)vergcs absolutciy: , then L:all converges, and
D I L; a S L;l a"l A serics converges conditionally if it
,,1 ,,1
III converges but not bsolt1!cly
~ ·(ompari ~m te\1 Assll m~ u".b,,>O
"11l1lI -If L;b" converges and either a,,£b,,(II ~ N) or a"lb"
has a limit, thell L;a" converges
-lfL;b" diverges and either b,,£a n (1I ~ N) or a,,Ibn has
a 110112('/'0 limit (or approaches 00), then L;a" diverges
The p-series and geometric scries are otien used tl"
comparisons Try a "limit" comparison when a series
looks like a p-series but is not directly comparable to it
E g L;SlIl (1 / n-) converges smee Hm , = 1
M
• Ratio & 1'00t tests Assume an" O
I
If }!" I· an or nl~ a tl , then L:u" converges
(ahsolutely) If ,Jim l ~ ~ >I()r ,!i!"lanl'/" >1 then
L;a" diverges These tests arc derived by comparison with
geometric serics The following are useful in applying the root test: ,!imn"' " = I (any p) and ,!im(II!)'" = = , More precisely ,!im ~(II!)"" = ~,
POWER SERIES
• Po\\er series A power series in x is a sequence of
\
polynomials inxofthc Ilmll L;a"x" (N=O.I 2, ",J,
,, "'· 0
The power series is denoted Lanx"
II n
A power series inx-c(or "centered at cO' or "about c") is written
L;a" (x-c)" =all +a, (x - c) +a, (x - c)' + "'
n ()
Replacing x with a real number q in a power series yields
a series of real numbers A power series converges at q if
the resulting series of real numbers converges
·Intenal ofcomerJ.:ence The set ufrealnu1l1bcrs at which
a power series converges is an interval , called the interval
of convergence or a point If the power series is centered
at c this set is either (i) (-00,00); (ii) (c-R , e+R) lor some
R>O possibly together with one or both endpoints; ur (iii) the point (' alonc In case (ii), R is called the radius of
convergence 01' the power series, which may be 00 and 0 t(,r cases (i) and (ii i) respectively Convergence is absolute for Ix-cl < R You can often determine a radius of convergence by solving the inequality that puts the ratio (or root) test limit less than I E.g., for
~ x " , I xl" l \ 2"n'! _ lxl ,
'~ -) , Iun - -, - " -1-1-" = - 2 <1=>l x l<2
which with the ratio test shows that the radius of
convergence is 2
-Geometric po\\"er \eries, A power series determines a
x • f(x)= L;a" (x-c)", One says the serie converges
" I)
to the tl1l1et,on The series L;x", i.e., the sequence of
" u
polynomials L;x" =1 +X+X2+ +X\= ·~(x *1)
converges for x in the interval (-1, I) to 1/( I-x) and diverges otherwise That is L; x" = - 1 1 (lx l<l) Other
geometric series may be identified through this basic one
E.g., L;2·;{ "x" = 2~i=(~)"=2x _ ~, _ for
" II 3" t) 3 3 1-x / {
Ix/31< I The interval of convergence is ( 3,3)
·(alculus or po\\er series Consider a function given by a
f ( x) = L;(I" t.t:-c)" ,
" (I
Such a function ·s differentiahle on (e-R , c+R) and its derivativc there is f' (x)= L;na" ( x-c)" "
" ,
The differentiated series has radius of convergence R, but
Such a tl111etion is integrable on (e - R e + R) and its
integral vanishing at cis:
f / W dt= L; _ a+ -" I(x - cY' ' (ix-cl<R)
4
The integrated series has radius of convcrgence R and may
E" _1_."'., l+x =I-x+x2"'illll,li"cs_1l+x _ =I-l:+t ·2 ,, The initial (geometric) series converges on (-1.1) and the integrated serics converges on (1,-1) The integration sas
In(I+x)=L;(-J)" '~' for Ixl<1; a remainder
" ,
argument (see below) implies equality for x=1
• Taylor and \lac! durin 'ric The Taylor serie about e of an infinitely diiTercntiable functionIis
L; T ( x - c lh= f(c)+f'(c)(x- e )+ T ( x - c)' + ·
If e=O it is also c lled a Maclaurin series The Ta)lor
converges to fix) if the remainder in Taylor's f(mnu la
R " (x)= (//+1l!'" I - {'"III('')·(X - c)''' 11 (I: ., b tween c and \' .,
1; varying with x and II) , approaches 0 as " 00 E.g the remainders at x=1 tllr the Maelaurin polynomials ot In( I +x) (in Taylor's formula above) satisfy
I R,,(l)I= 1 ' <; n+11 .()
(11+ 1)(1+;)"
so In2= L; - -n - - '
" ,
·Computinl! 11~I r If R>O and
f(x)= L;a,,(x - c)"(lx - cl<R) then the wen'icients are
11 II
necessarily the Taylor coetficients: Q ,, = f''''(l')/n! This
means Taylor series may be found other than by direetlv
series gives 1_ , = L;nx" ' = L;(// + llx" (Ix i< n
• Ba ic I\l:tcLaurin wri,
_ I -1_= x I+x+\,2+ , = 11 L;x" II (lxl<l )
arct a nx=x~ ~+~ - "' = L: ( - J)1/ ~ ~" I (lxj'5
,{ ~ " " 2//+
The li)lIowing hold tor all real x:
x~ x:l " " x"
e " = I + x + ?T+ ",+"'= L i
x:! Xl ( - 1 ) "x:!rI
cos x = l-x+?T+-4t - " '= L; -(- ,) -)- ,
II II _n
• x: i x!i (-1)"x:.!"
SInx=x - -;- l,+~,-"' L; = (? +
• Binomial \crie For P" O and ti" Ixl < I
(l+x)I ' =I+px + - - ? - , - x'+ ,= L; x·,
(]=l C)=p, ( J
The binomial coefficients are
(")= P(p-1)(p-2!'" (p-k+
Ifp is a positive integer, C~O Illr k > p
A ri g ht!l "' ·_ ""rI CII 'f ' par t 01 till' Jlublica rio!l rn.l~ b : rerro.lu('cu or
l(':1n~l1l1t\o:dl!\nn fom l ,orInJn)~,
clcdr'(lI1IC or II1cchalm:al n1<:ILlJIn~
[lhuIOCOP).rcc:orJUll:,l.oran\ InfOlll\lI1tl<n
~tOr.lgC an d rClnl"\'31 s~)I("m Uhvul
\HllI e n rcrnli"'10n frolll the rubh.JlcT f\ 2Ui.1I - Z00 7 UnCh ll r U ln c 0 08
~u lt , : Due I U l IS condensed
fqfTnQI, plc,uc II!>C thl
\ )uid.)tudv a~ II 111,111.1 bUl nOI h a rep l acement fo r
" " l gnc J ,·l t$S\\ ' Ir\;
Customer Hotline # 1,800,230,9522
ISBN-13' 978-157222475-9 ISBN - 1D: 157222475-4
9 11 ~ ll)llli ~1I1!1!1!IJIJ ~ l llrlllllilllllll