Handbook of mathematics for engineers and scienteists part 165 pptx

1116 FINITESUMS ANDINFINITESERIES2... Sums involving trigonometric functions... 1118 FINITESUMS ANDINFINITESERIES11... Other numerical series.1... 1120 FINITESUMS ANDINFINITESERIES16...

1116 FINITESUMS ANDINFINITESERIES

2

n



k=1

sin2m πk

2n = n

22m C2m m+ 1

2, m<2n. 3

n–1



k=0

(–1)kcosm πk

n = 1 2



1– (–1)m+n

, m=0,1, , n –1

4

n–1



k=0

(–1)kcosn πk

n = n

2n–1.

T1.1.2 Functional Sums

T1.1.2-1 Sums involving hyperbolic functions

1

n–1



k=0

sinh(kx + a) = sinh



n–1

2 x + a



sinh(nx/2)

sinh(x/2) . 2

n–1



k=0

cosh(kx + a) = cosh



n–1

2 x + a



sinh(nx/2)

sinh(x/2) . 3

n–1



k=0

(–1)k sinh(kx + a) = 1

2cosh(x/2)

 sinh



a– x 2

 + (–1)nsinh

2

n–1

2 x + a



4

n–1



k=0

(–1)k cosh(kx + a) = 1

2cosh(x/2)

 cosh



a– x 2

 + (–1)ncosh

2

n–1

2 x + a



5

n–1



k=1

k sinh(kx + a) = – 1

sinh2(x/2)



n sinh[(n –1)x + a] – (n –1) sinh(nx + a) – sinh a

4

6

n–1



k=1

k cosh(kx + a) = – 1

sinh2(x/2)



n cosh[(n –1)x + a] – (n –1) cosh(nx + a) – cosh a

4

7

n–1



k=1

(–1)k k sinh(kx + a) = 1

cosh2(x/2)

 (–1)n–1n sinh[(n –1)x + a]

+ (–1)n–1(n –1) sinh(nx + a) – sinh a

4 8

n–1



k=1

(–1)k k cosh(kx + a) = 1

cosh2(x/2)

 (–1)n–1n cosh[(n –1)x + a]

+ (–1)n–1(n –1) cosh(nx + a) – cosh a

4 9

n



k=0

C k

n sinh(kx + a) =2ncoshn x

2 sinh



nx

2 + a



10

n



k=0

C k

n cosh(kx + a) =2ncoshn x

2 cosh



nx

2 + a



n–1



k=1

a k sinh(kx) = a sinh x – a n sinh(nx) + a n+1sinh[(n –1)x]

1–2a cosh x + a2 .

12

n–1



k=0

a k cosh(kx) = 1– a cosh x – a n cosh(nx) + a n+1cosh[(n –1)x]

1–2a cosh x + a2 .

13

n



k=1

1

2k tanh

x

2k = coth x –

1

2n coth

x

2n.

14

n–1



k=0

2ktanh(2k x) =2ncoth(2n x ) – coth x.

T1.1.2-2 Sums involving trigonometric functions

1

n



k=1

sin(2kx ) = sin[(n +1)x] sin(nx) cosec x.

2

n



k=0

cos(2kx ) = sin[(n +1)x] cos(nx) cosec x.

3

n



k=1

sin[(2k–1)x] = sin2(nx) cosec x.

4

n



k=1

cos[(2k–1)x] = sin(nx) cos(nx) cosec x.

5

n–1



k=0

sin(kx + a) = sin



n–1

2 x + a

 sin nx

2 cosec

x

2. 6

n–1



k=0

cos(kx + a) = cos



n–1

2 x + a

 sin nx

2 cosec

x

2. 7

2n–1

k=0

(–1)k cos(kx + a) = sin

2

n–1

2 x + a



sin(nx) sec x

2. 8

n



k=1

(–1)k+1sin[(2k–1)x] = (–1)n+1sin(2nx)

2cos x .

9

n



k=1

(–1)kcos(2kx) = –1

2 + (–1)n

cos[(2n+1)x]

2cos x .

10

n



k=1

sin2(kx) = n

2 –

cos[(n +1)x] sin(nx)

2sin x .

1118 FINITESUMS ANDINFINITESERIES

11

n



k=1

cos2(kx) = n

2 +

cos[(n +1)x] sin(nx)

2sin x .

12

n–1



k=1

ksin(2kx) = sin(2nx)

4sin2x – ncos[(2n–1)x]

2sin x .

13

n–1



k=1

kcos(2kx) = nsin[(2n–1)x]

2sin x –

1– cos(2nx)

4sin2x 14

n–1



k=1

a k sin(kx) = a sin x – a n sin(nx) + a n+1sin[(n –1)x]

1–2a cos x + a2 .

15

n–1



k=0

a k cos(kx) = 1– a cos x – a n cos(nx) + a n+1cos[(n –1)x]

1–2a cos x + a2 .

16

n



k=0

C k

n sin(kx + a) =2ncosn x

2 sin



nx

2 + a



17

n



k=0

C k

n cos(kx + a) =2ncosn x

2 cos



nx

2 + a



18

n



k=0

(–1)k C k

n sin(kx + a) = (–2)nsinn x

2 sin



nx

2 +

πn

2 + a



19

n



k=0

(–1)k C k

n cos(kx + a) = (–2)nsinn x

2 cos



nx

2 +

πn

2 + a



20

n



k=1



2ksin2 x

2k

2

=



2nsin2 x

2n

2 – sin2x

21

n



k=0

1

2k tan

x

2k =

1

2n cot

x

2n –2cot(2x)

T1.2 Infinite Series

T1.2.1 Numerical Series

T1.2.1-1 Progressions

1

∞



k=0

aq k = a

1– q, |q|<1 2

∞



k=0

(a + bk)q k= a

1– q +

bq

(1– q)2, |q|<1

T1.2.1-2 Other numerical series.

1

∞



n=0

(–1)n

n+1 = ln2.

2

∞



n=0

(–1)n

2n+1 =

π

4. 3

∞



n=1

1

n (n +1) =1

4

∞



n=1

(–1)n

n (n +1) =1–2ln2

5

∞



n=1

1

n (n +2) =

3

4. 6

∞



n=1

(–1)n

n (n +2) = –

1

4. 7

∞



n=1

1 (2n–1)(2n+1) =

1

2. 8

∞



n=1

1

n2 =

π2

6 . 9

∞



n=1

(–1)n+1

n2 =

π2

12. 10

∞



n=1

1

(2n–1)2 =

π2

8 . 11

∞



n=1

1

n2+ a2 =

π

2a coth(πa) – 1

2a2. 12

∞



n=1

1

n2– a2 = –

π

2a cot(πa) + 1

2a2. 13

∞



k=1

1

k2n =

22n–1π2n

(2n)! |B2n|; the B2nare Bernoulli numbers

14

∞



k=1

(–1)k+1

k2n =

(22n–1–1)π2n

(2n)! |B2n|; the B2nare Bernoulli numbers

15

∞



k=1

1

(2k–1)2n =

(22n–1–1)π2n

2(2n)! |B2n|; the B2nare Bernoulli numbers

1120 FINITESUMS ANDINFINITESERIES

16

∞



k=1

1

k2k = ln2

17

∞



k=0

(–1)k

n2k =

n2

n2+1. 18

∞



k=0

1

k! = e =2.71828 .

19

∞



k=0

(–1)k

k! =

1

e =0.36787 .

20

∞



k=1

k

(k +1)! =1

T1.2.2 Functional Series

T1.2.2-1 Power series

1

∞



k=0

x k= 1

1– x, |x|<1

2

∞



k=1

kx k= x

(1– x)2, |x|<1 3

∞



k=1

k2x k= x (x +1)

(1– x)3 , |x|<1 4

∞



k=1

k3x k= x(1+4x + x2)

(1– x)4 , |x|<1 5

∞



k=0

( 1)k k n x k=

x d dx

n 1

1x, |x|<1 6

∞



k=1

x k

k = – ln(1– x), –1 ≤x<1

7

∞



k=1

(–1)k–1x k

k = ln(1+ x), |x|<1 8

∞



k=1

x2k–1

2k–1 =

1

2 ln

1+ x

1– x, |x|<1 9

∞



k=1

(–1)k–1 x

2k–1

2k–1 = arctan x, |x| ≤ 1.

∞



k=1

x k

k2 = –

 x 0

ln(1– t)

t dt, |x| ≤ 1 11

∞



k=1

x k+1

k (k +1) = x + (1– x) ln(1– x), |x| ≤ 1

12

∞



k=1

x k+2

k (k +2) =

x

2 +

x2

4 +

1

2(1– x2) ln(1– x), |x| ≤ 1. 13

∞



k=0

x k

k! = e

x, xis any number.

14

∞



k=0

x2k

(2k)! = cosh x, xis any number.

15

∞



k=0

(–1)k x

2k

(2k)! = cos x, xis any number.

16

∞



k=0

x2k+1

(2k+1)! = sinh x, xis any number.

17

∞



k=0

(–1)k x

2k+1 (2k+1)! = sin x, xis any number.

18

∞



k=0

x k+1

k ! (k +1) = e

x–1, xis any number

19

∞



k=0

x k+2

k ! (k +2) = (x –1)e x+1, xis any number

20

∞



k=0

(–1)k x

2k+1

k! (2k+1) =

√ π

2 erf x, xis any number.

21

∞



k=0

(k + a) n

k! x

k=

d n

dt n exp(at + xe t



t=0, xis any number.

22

∞



k=1

22k(22k–1)|B2k|

(2k)! x

2k–1= tan x; the B

2kare Bernoulli numbers, |x|< π/2

23

∞



k=1

(–1)k–122k(22k–1)|B2k|

(2k)! x

2k–1= tanh x; the B

2kare Bernoulli numbers,|x|< π/2

24

∞



k=1

22k|B2k|

(2k)! x

2k–1= 1

x – cot x; the B2kare Bernoulli numbers, 0<|x|< π.

25

∞



k=1

(–1)k–122k|B2k|

(2k)! x

2k–1= coth x – 1

x; the B2kare Bernoulli numbers, |x|< π.

1122 FINITESUMS ANDINFINITESERIES

T1.2.2-2 Trigonometric series in one variable involving sine

1

∞



k=1

1

k sin(kx) = 1

2(π – x), 0< x <2π. 2

∞



k=1

(–1)k–1

k sin(kx) = 1

2x, –π < x < π.

3

∞



k=1

a k

k sin(kx) = arctan a sin x

1– a cos x, 0< x <2π, |a| ≤ 1 4

∞



k=0

1

2k+1 sin(kx) =

π

4 cos

x

2 – sin

x

2 ln

 cot2 x 4

 , 0< x <2π

5

∞



k=0

(–1)k

2k+1 sin(kx) = –

1

4 cos

x

2 ln

 cot2 x + π 4

 – π

4 sin

x

2, –π < x < π.

6

∞



k=1

1

k2 sin(kx) = –

 x

0 ln



2sin t 2



dt, 0 ≤x < π.

7

∞



k=1

(–1)k

k2 sin(kx) = –

 x

0 ln



2cos t 2



dt, –π < x < π.

8

∞



k=1

1

k (k +1) sin(kx) = (π – x) sin

2 x

2 + sin x ln



2sin x 2

 , 0 ≤x≤ 2π

9

∞



k=1

(–1)k

k (k +1) sin(kx) = –x cos

2 x

2 + sin x ln



2cosx 2

 , –π ≤x≤π

10

∞



k=1

k

k2+ a2 sin(kx) =

π

2sinh(πa) sinh[a(π – x)], 0< x <2π 11

∞



k=1

(–1)k+1 k

k2+ a2 sin(kx) =

π

2sinh(πa) sinh(ax), –π < x < π.

12

∞



k=1

k

k2– a2 sin(kx) =

π

2sin(πa) sin[a(π – x)], 0< x <2π 13

∞



k=1

(–1)k+1 k

k2– a2 sin(kx) =

π

2sin(πa) sin(ax), –π < x < π.

14

∞



k=2

(–1)k k

k2–1 sin(kx) =

1

4 sin x +

1

2x cos x, –π < x < π.

15

∞



k=1

1

k2n+1 sin(kx) =

(–1)n–1(2π)2n+1

2(2n+1)! B2n+1



x

2π

 , where0 ≤x≤ 2π for n =1, 2, ; 0< x <2π for n =0; and the B n (x) are Bernoulli

polynomials