MATHEMATICAL METHOD IN SCIENCE AND ENGINEERING Episode 13 pdf

In the following tests we either consider series with positive terms or take the absolute value of the terms; hence we check for absolute convergence.. Comparison test: Consider the seri

CONVERGENCE TESTS 433

15.3 CONVERGENCE TESTS

There exist a number of tests for checking the convergence of a given series In what follows we give some of the most commonly used tests for convergence The tests are ordered in increasing level of complexity In practice one starts with the simplest test and, if the test fails, moves on to the next one In the following tests we either consider series with positive terms or take the absolute value of the terms; hence we check for absolute convergence

15.3.1 Comparison Test

The simplest test for convergence is the comparison test We compare a given series term by term with another series convergence or divergence of which has been established Let two series with the general terms a, and b, be given For all n 2 1 if la,l 5 Ib,l is true and if the series c,"==, lbnl is convergent, then the series Cr=p=I a, is also convergent Similarly, if xr=, a, is divergent, then the series C,"=l lbnl is also divergent

Example 15.3 Comparison test: Consider the series with the general term

a, = n-P where p = 0.999, We compare this series with the harmonic series which has the general term b, = n-' Since for n 2 1 we can write n-' < n-0.999 and since the harmonic series is divergent, we also conclude that the series xr=l n-p is divergent

15.3.3 Cauchy Root Test

For the series x,"=l a,, when we find the limit

lim = 1,

for 1 < 1 the series is convergent, for 1 > 1 the series is divergent, and for 1 = 1 the test is inconclusive

Let a, = f (n) be the general term of a given series with positive terms If

for n > 1, f (n) is continuous and a monotonic decreasing function, that

is, f (n+ 1) < f (n), then the series converges or diverges with the integral

From here it is apparent that in the limit as N -+ co, if the integral

J: f (x) dx is finite, then the series CT=l an is convergent If the inte- gral diverges, then the series also diverges

Example 15.4 Integral test: Let us consider the Riemann zeta function

CONVERGENCE TESTS 435

-f 1; thus the ratio test fails

an+ 1

In the limit as n + this gives - an

However, using the integral test we find

m = 1 the Raabe test is inconclusive

The Raabe test can also be expressed as follows: Let N be a positive

integer independent of n For all n 2 N , if n (e - 1) 2 P > 1 is true, then the series is convergent and if n (e - 1) 5 1 is true, then the series

Example 15.6 Raabe test: The second form of the Raabe test shows that

is divergent This follows from the fact that the harmonic series C,"xl

for all n values,

(15.17) When the available tests fail, we can also use theorems like the Cauehy theorem

436 INFINITE SERIES

Example 15.7 Cuwhy theorem; Let us check the convergence of the se- ries

1 nlna n'

03

1 +- 1 f T + ' = 1 c-

n= 2

by using the Cauchy theorem for (Y 2 0 Choosing the value of c as two,

we construct the series xr=l 2 " ~ " = 2a2 + 4a4 + 8as + , where the general term is given as

(15.20) Since the series C,"=* -& converges for a > 1, our series is also convergent for (Y > 1 On the other hand, for (Y 5 1, both series are divergent

15.3.7

Legendre series are given as

Gauss Test and Legendre Series

Cn=Oa2nz2n 00 and C~=oa2n+lz2n+1, 3: E [-I, 11 (15.21) Both series have the same recursion relation

, n=O,1,

( n - I) (I + n + 1) (n + 1) (n + 2)

For 121 < 1, convergence of both series can be established by using the ratio test For the even series the general term is given as un = a2nz2n; hence we write

(an - I ) (an + I + 1) z 2

Un QnX2n (an + 1) (an + 2 ) '

- - un+1 - a2n+1x2n+1 - (15.23)

(15.24) Using the ratio test we conclude that the Legendre series with the even terms

is convergent for the interval z E (-1,l) The argument and the conclusion for the other series are exactly the same However, at the end points the ratio test fails For these points we can use the Gauss test:

Gauss test:

Let C,"==, u, be a series with positive terms If for n 2 N ( N is a given constant) we can write

(15.25)

CONVERGENCE TESTS 437

where 0 (5) means that for a given function f (n) thelimit limn+OO{.f (n) /$}

is finite, then the C,"==, un series converges for p > 1 and diverges for p 5 1

Note that there is no case here where the test fails

Example 15.8 Legendre series: We now investigate the convergence of the Legendre series a t the end points, z = fl, by using the Gauss test

We find the required ratio as

n+m [4n2 + 2n - 1 ( I + l)] n " 2 ) = 4 '

we see that this ratio is constant and goes as O ( 3 ) Since p = 1 in

-, we conclude that the Legendre series (both the even and the odd series) diverge at the end points

Un

un+ 1

Example 15.9 Chebyshev series: The Chebyshev equation is given as

d2Y dY (1 - 2)- - z- + n2y = 0

d x 2 dx (15.29)

Let us find finite solutions of this equation in the interval z E [-1,1] by using the Frobenius method We substitute the following series and its derivatives into the Chebyshev equation:

This gives the indicial equation as

ALGEBRA OF SERlES 439

This gives us the limit

Using the ratio test it is clear that this series converges for the interval (-1,l) However, a t the end points the ratio test fails, where we now use the Raabe test We first evaluate the ratio

(15.43)

= lim k [ ] = 5 > 1 , (15.44)

- l1

(2k + 2)(2lc + 1) lim k - - 1 = lim k

k - c c [ U z t t 2 ] k+oo [ (2k)2-n2

6 k + 2 + n 2 3

k - c o (21C)Z - n2

which indicates that the series is convergent a t the end points as well

This means that for the polynomial solutions of the Chebyshev equation, restricting n to integer values is an additional assumption, which is not required by the finite solution condition at the end points The same conclusion is valid for the series with the odd powers

15.3.8 Alternating Series

For a given series of the form Cr=’=, (-l),+’ a,, if a, is positive for all n, then the series is called an alternating series In an alternating series for sufficiently large values of n, if a, is monotonic decreasing or constant and the limit

lim a, = 0

is true, then the series is convergent This is also known as the Leibniz rule

Example 15.10 Leibnix rule: In the alternating series

since $ > 0 and $ -+ 0 as n -+ 00, the series is convergent

15.4 ALGEBRA OF SERIES

Absolute convergence is very important in working with series It is only for absolutely convergent series that ordinary algebraic manipulations (addition, subtraction, multiplication, etc.) can be done without problems:

1 An absolutely convergent series can be rearranged without affecting the sum

440 INFINITE SERlES

2 Two absolutely convergent series can be multiplied with each other The result is another absolutely convergent series, which converges to the multiplication of the individual series sums

All these operations look very natural; however, when applied to condi- tionally convergent series they may lead to erroneous results

Example 15.11 Conditionally convergent series: The following condi- tionally convergent series:

= 1 - (- - -) - (- - -) - (15.47)

= 1-0.167-0.05-. , (15.48) obviously converges to some number less than one, actually to In 2 = 0.693 We now rearrange this sum as

~g = 1.5078,

It is now seen that this alternating series added in this order converges

to 3/2 What we have done is very simple First we added positive terms until the partial sum was equal or just above 3/2 and then subtracted negative terms until the partial sum fell just below 3/2 In this process

we have neither added nor subtracted anything from the series; we have simply added its terms in a different order

By a suitable arrangement of its terms a conditionally convergent series can be made to converge to any desired value or even to diverge This result is also known as the Riemann theorem

15.4.1 Rearrangement of Series

Let us write the partial sum of a double series as

(15.51)

and say that the double series CG=, aij is convergent and has the sum s

When a double sum

Writing both sums explicitly we get

(15.56)

(15.57)

Another rearrangement can be obtained by the definitions

a00 + a01 + a02 + +alo + ao3 + all +

USEFUL INEQUALITIES ABOUT SERIES

Let + = 1; then we can state the following useful inequalities about series:

H8lder’s Inequality: If an 2 0, b, 2 0, p > 1, then

cx)

c anbn 5 (2 u;) I ” (g b;) ‘ I q (15.61)

Minkowski’s Inequality: If an 2.0, b, 2 0 and p 2 1, then

[ n= 2 1 (an + ha)’] I (2 n=l a;) + (5 n=l K) (15.62)

Schwarz-Cauchy Inequality: If a, 2 0, and bn 2 0, then

SERIES OF FUNCTIONS 443

fig 15.2 Uniform convergence is very import.ant

In studying the properties of series of functions we need a new concept called the uniform convergence

15.6.1 Uniform Convergence

For a given positive small number E , if there exists a number N independent

of z for z E [a,b], and if for all n 2 N we can say the inequality

I.(.) - .%&)I < E (15.66)

is true, then the series with the general term un(z) is uniformly convergent in the interval [a, b] This also means that for a uniformly convergent series and for a given error margin E , we can always find N number of terms independent

of z such that when added the remainder of the series, that is

(15.67)

is always less than E for all z in the interval [a, b] Uniform convergence can also be shown as in Figure 15.2

15.6.2 Weierstrass M-Test

For uniform convergence the most commonly used test is the Weierstrass

M or in short the M-test: Let us say that we found a series of numbers

444 INFINITE SERIES

xzl Mi, such that for all 3: in [a, 61 the inequality M; 2 Iui(x)l is true Then the uniform convergence of the series of functions czl ui(z), in the interval [a, b] , follows from the convergence of the series of numbers xzl Mi Note that because the absolute values of ui(z) are taken, the M-test also checks absolute convergence However, it should be noted that absolute convergence and uniform convergence are two independent concepts and neither of them implies the other

Example 15.12 M-test: The following series are uniformly convergent, but not absolutely convergent:

(15.68)

while the series (the so-called Riemann zeta function)

(15.69) converges uniformly and absolutely in the interval [a,m), where a is any number greater than one Because the M-test checks for uniform and absolute convergence together, for conditionally convergent series

we can use the Abel test

15.6.3 Abel Test

Let a series with the general term u,(z) = a,f,(z) be given If the series

of numbers X u , = A is convergent and if the functions f,(x) are bounded,

0 5 f,(x) 5 M , and monotonic decreasing, fn+l(z) 5 f,(x), in the interval

[a, b] , then the series C u,(z) is uniformly convergent in [a, 61

Example 15.13 Unaform convergence: The series

is absolutely convergent but not uniformly convergent in [0,1] From the definition of uniform convergence it is clear that any series

TAYLOR SERIES 445

15.6.4

For a uniformly convergent series the following are true:

Properties of Uniformly Convergent Series

1 If u,(z) for all n are continuous, then the series

where the integral sign can be interchanged with the summation sign

3 If for all n in the interval [a, b] , ~ ( x ) and -&(x) are continuous, and the series C,"==, gun(.) is uniformly convergent, then we can differen- tiate the series term by term as

(15.74) (15.75)

is called the remainder, and it can also be written as

Note that Equation (15.79) is exact When the limit

15.7.1 Maclaurin Theorem

In the Taylor series if we take the point of expansion as the origin, we obtain the Maclaurin series:

(15.84)

(15.87) X"

(1 +x)" -

n=O

- n=O C n!(m - n)!

It can be easily shown that this series is convergent in the interval -1 < x <

1 Note that for m = n (integer) the sum automatically terminates after a finite number of terms, where the quantity (z) = m!/n!(m - n)! is called the binomial coefficient

Example 15.14 Relativistic kinetic energy: The binomial formula is prob- ably one of the most widely used formulas in science and engineering

An important application of the binomial formula was given by Einstein

in his celebrated paper where he announced his famous formula for the energy of a freely moving particle of mass m as

1101~20 , I n 0

(15.97)

f (Xl,X2, "', Z m ) = n!

n=o

where the coefficients a, are independent of x To use the ratio test we write

and find the limit

(15.99)

(15.100) Hence the condition for the convergence of a power series is obtained as

1x1 < R*-R< x < R, ( 15.10 1) where R is called the radius of convergence At the end points the ratio test fails; hence these points must be analyzed separately

Example 15.14 Power series: For the power series

Example 15.15 Power series: The radius of convergence can also be zero For the series

1 + x + 2!x2 + 3!x3 + + n!xn + , (15.103) the ratio

gives

1 lim (n + 1) = -

n+m R - ) O 0

(15.104)

(15.105) Thus the radius of convergence is zero Note that this series converges only for x = 0

15.8.1 Convergence of Power Series

If a power series is convergent in the interval -R < x < R, then it is uniformly and absolutely convergent in any subinterval S:

-S 5 x 5 S, where 0 < S < R (15.109) This can be seen by taking Mi = /ail Sz in the M-test

15.8.3

In the interval of uniform convergence a power series can be differentiated and integrated as often as desired These operations do not change the radius of convergence

Differentiation and Integration of Power Series

then b, = a, is true for all n Hence the power series is unique

Proof: Let us write

m oc

c anxn = b,xn 3 -R < x < R,

n=O n=O

where R is equal to the smaller of the two radii R, and R b If we set

x = 0 in this equation we find a0 = bo Using the fact that a power series can be differentiated as often as desired, we differentiate the above equation once t o write

15.8.5 Inversion of Power Series

Consider the power series expansion of the function y(x) - yo in powers of

( 2 - 20) as

y - yo = a1 (x - 20) + aa(z - 20) 2 + , (15.11 1) that is,

15.9 SUMMATION OF INFINITE SERIES

After we conclude that a given series is convergent, the next and most impor- tant thing we need in applications is the value or the function that it converges

to For uniformly convergent series it is sometimes possible to identify an un- known series as the derivatives or the integrals of a known series In this section we introduce some analytic techniques to evaluate the sums of infinite series We start with the Euler-Maclaurin sum formula, which has important applications in quantum field theory and Green’s function calculations Next

we discuss how some infinite series can be summed by using the residue t h e orem Finally, we show that differintegrals can also be used to sum infinite series