Handbook of mathematics for engineers and scienteists part 165 pptx

1116 FINITE SUMS AND INFINITE SERIES 2. n  k=1 sin 2m πk 2n = n 2 2m C m 2m + 1 2 , m < 2n. 3. n–1  k=0 (–1) k cos m πk n = 1 2  1 –(–1) m+n  , m = 0, 1, , n – 1. 4. n–1  k=0 (–1) k cos n πk n = n 2 n–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 2 cosh(x/2)  sinh  a – x 2  +(–1) n sinh  2n – 1 2 x + a  . 4. n–1  k=0 (–1) k cosh(kx + a)= 1 2 cosh(x/2)  cosh  a – x 2  +(–1) n cosh  2n – 1 2 x + a  . 5. n–1  k=1 k sinh(kx + a)=– 1 sinh 2 (x/2)  n sinh[(n – 1)x + a]–(n – 1)sinh(nx + a)–sinha  . 6. n–1  k=1 k cosh(kx+a)=– 1 sinh 2 (x/2)  n cosh[(n –1)x+a]–(n –1)cosh(nx +a)–cosha  . 7. n–1  k=1 (–1) k k sinh(kx + a)= 1 cosh 2 (x/2)  (–1) n–1 n sinh[(n – 1)x + a] +(–1) n–1 (n – 1)sinh(nx + a)–sinha  . 8. n–1  k=1 (–1) k k cosh(kx + a)= 1 cosh 2 (x/2)  (–1) n–1 n cosh[(n – 1)x + a] +(–1) n–1 (n – 1)cosh(nx + a)–cosha  . 9. n  k=0 C k n sinh(kx + a)=2 n cosh n x 2 sinh  nx 2 + a  . 10. n  k=0 C k n cosh(kx + a)=2 n cosh n x 2 cosh  nx 2 + a  . T1.1. FINITE SUMS 1117 11. n–1  k=1 a k sinh(kx)= a sinh x – a n sinh(nx)+a n+1 sinh[(n – 1)x] 1 – 2a cosh x + a 2 . 12. n–1  k=0 a k cosh(kx)= 1 – a cosh x – a n cosh(nx)+a n+1 cosh[(n – 1)x] 1 – 2a cosh x + a 2 . 13. n  k=1 1 2 k tanh x 2 k =cothx – 1 2 n coth x 2 n . 14. n–1  k=0 2 k tanh(2 k x)=2 n coth(2 n x)–cothx. T1.1.2-2. Sums involving trigonometric functions. 1. n  k=1 sin(2kx)=sin[(n + 1)x]sin(nx)cosecx. 2. n  k=0 cos(2kx)=sin[(n + 1)x]cos(nx)cosecx. 3. n  k=1 sin[(2k – 1)x]=sin 2 (nx)cosecx. 4. n  k=1 cos[(2k – 1)x]=sin(nx)cos(nx)cosecx. 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  2n – 1 2 x + a  sin(nx)sec x 2 . 8. n  k=1 (–1) k+1 sin[(2k – 1)x]=(–1) n+1 sin(2nx) 2 cos x . 9. n  k=1 (–1) k cos(2kx)=– 1 2 +(–1) n cos[(2n + 1)x] 2 cos x . 10. n  k=1 sin 2 (kx)= n 2 – cos[(n + 1)x]sin(nx) 2 sin x . 1118 FINITE SUMS AND INFINITE SERIES 11. n  k=1 cos 2 (kx)= n 2 + cos[(n + 1)x]sin(nx) 2 sin x . 12. n–1  k=1 k sin(2kx)= sin(2nx) 4 sin 2 x – n cos[(2n – 1)x] 2 sin x . 13. n–1  k=1 k cos(2kx)= n sin[(2n – 1)x] 2 sin x – 1 –cos(2nx) 4 sin 2 x . 14. n–1  k=1 a k sin(kx)= a sin x – a n sin(nx)+a n+1 sin[(n – 1)x] 1 – 2a cos x + a 2 . 15. n–1  k=0 a k cos(kx)= 1 – a cos x – a n cos(nx)+a n+1 cos[(n – 1)x] 1 – 2a cos x + a 2 . 16. n  k=0 C k n sin(kx + a)=2 n cos n x 2 sin  nx 2 + a  . 17. n  k=0 C k n cos(kx + a)=2 n cos n x 2 cos  nx 2 + a  . 18. n  k=0 (–1) k C k n sin(kx + a)=(–2) n sin n x 2 sin  nx 2 + πn 2 + a  . 19. n  k=0 (–1) k C k n cos(kx + a)=(–2) n sin n x 2 cos  nx 2 + πn 2 + a  . 20. n  k=1  2 k sin 2 x 2 k  2 =  2 n sin 2 x 2 n  2 –sin 2 x. 21. n  k=0 1 2 k tan x 2 k = 1 2 n cot x 2 n – 2 cot(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. INFINITE SERIES 1119 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 – 2 ln 2. 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 n 2 = π 2 6 . 9. ∞  n=1 (–1) n+1 n 2 = π 2 12 . 10. ∞  n=1 1 (2n – 1) 2 = π 2 8 . 11. ∞  n=1 1 n 2 + a 2 = π 2a coth(πa)– 1 2a 2 . 12. ∞  n=1 1 n 2 – a 2 =– π 2a cot(πa)+ 1 2a 2 . 13. ∞  k=1 1 k 2n = 2 2n–1 π 2n (2n)! |B 2n |;theB 2n are Bernoulli numbers. 14. ∞  k=1 (–1) k+1 k 2n = (2 2n–1 – 1)π 2n (2n)! |B 2n |;theB 2n are Bernoulli numbers. 15. ∞  k=1 1 (2k – 1) 2n = (2 2n–1 – 1)π 2n 2(2n)! |B 2n |;theB 2n are Bernoulli numbers. 1120 FINITE SUMS AND INFINITE SERIES 16. ∞  k=1 1 k2 k =ln2. 17. ∞  k=0 (–1) k n 2k = n 2 n 2 + 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 k 2 x k = x(x + 1) (1 – x) 3 , |x| < 1. 4. ∞  k=1 k 3 x k = x(1 + 4x + x 2 ) (1 – x) 4 , |x| < 1. 5. ∞  k=0 ( 1) k k n x k =  x d dx  n 1 1 x , |x| < 1. 6. ∞  k=1 x k k =–ln(1 – x), –1 ≤ x < 1. 7. ∞  k=1 (–1) k–1 x k k =ln(1 + x), |x| < 1. 8. ∞  k=1 x 2k–1 2k – 1 = 1 2 ln 1 + x 1 – x , |x| < 1. 9. ∞  k=1 (–1) k–1 x 2k–1 2k – 1 =arctanx, |x| ≤ 1. T1.2. INFINITE SERIES 1121 10. ∞  k=1 x k k 2 =–  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 + x 2 4 + 1 2 (1 – x 2 )ln(1 – x), |x| ≤ 1. 13. ∞  k=0 x k k! = e x , x is any number. 14. ∞  k=0 x 2k (2k)! =coshx, x is any number. 15. ∞  k=0 (–1) k x 2k (2k)! =cosx, x is any number. 16. ∞  k=0 x 2k+1 (2k + 1)! =sinhx, x is any number. 17. ∞  k=0 (–1) k x 2k+1 (2k + 1)! =sinx, x is any number. 18. ∞  k=0 x k+1 k!(k + 1) = e x – 1, x is any number. 19. ∞  k=0 x k+2 k!(k + 2) =(x – 1)e x + 1, x is any number. 20. ∞  k=0 (–1) k x 2k+1 k!(2k + 1) = √ π 2 erf x, x is any number. 21. ∞  k=0 (k + a) n k! x k =  d n dt n exp(at + xe t )  t=0 , x is any number. 22. ∞  k=1 2 2k (2 2k – 1)|B 2k | (2k)! x 2k–1 =tanx;theB 2k are Bernoulli numbers, |x| < π/2. 23. ∞  k=1 (–1) k–1 2 2k (2 2k – 1)|B 2k | (2k)! x 2k–1 =tanhx;theB 2k are Bernoulli numbers, |x| < π/2. 24. ∞  k=1 2 2k |B 2k | (2k)! x 2k–1 = 1 x –cotx;theB 2k are Bernoulli numbers, 0 < |x| < π. 25. ∞  k=1 (–1) k–1 2 2k |B 2k | (2k)! x 2k–1 =cothx – 1 x ;theB 2k are Bernoulli numbers, |x| < π. 1122 FINITE SUMS AND INFINITE SERIES 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 2 x,–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  cot 2 x 4  , 0 < x < 2π. 5. ∞  k=0 (–1) k 2k + 1 sin(kx)=– 1 4 cos x 2 ln  cot 2 x + π 4  – π 4 sin x 2 ,–π < x < π. 6. ∞  k=1 1 k 2 sin(kx)=–  x 0 ln  2 sin t 2  dt, 0 ≤ x < π. 7. ∞  k=1 (–1) k k 2 sin(kx)=–  x 0 ln  2 cos t 2  dt,–π < x < π. 8. ∞  k=1 1 k(k + 1) sin(kx)=(π – x)sin 2 x 2 +sinx ln  2 sin x 2  , 0 ≤ x ≤ 2π. 9. ∞  k=1 (–1) k k(k + 1) sin(kx)=–x cos 2 x 2 +sinx ln  2 cos x 2  ,–π ≤ x ≤ π. 10. ∞  k=1 k k 2 + a 2 sin(kx)= π 2 sinh(πa) sinh[a(π – x)], 0 < x < 2π. 11. ∞  k=1 (–1) k+1 k k 2 + a 2 sin(kx)= π 2 sinh(πa) sinh(ax), –π < x < π. 12. ∞  k=1 k k 2 – a 2 sin(kx)= π 2 sin(πa) sin[a(π – x)], 0 < x < 2π. 13. ∞  k=1 (–1) k+1 k k 2 – a 2 sin(kx)= π 2 sin(πa) sin(ax), –π < x < π. 14. ∞  k=2 (–1) k k k 2 – 1 sin(kx)= 1 4 sin x + 1 2 x cos x,–π < x < π. 15. ∞  k=1 1 k 2n+1 sin(kx)= (–1) n–1 (2π) 2n+1 2(2n + 1)! B 2n+1  x 2π  , where 0 ≤ x ≤ 2π for n = 1, 2, ; 0 < x < 2π for n = 0;andtheB n (x) are Bernoulli polynomials. . π. 15. ∞  k=1 1 k 2n+1 sin(kx)= (–1) n–1 (2π) 2n+1 2(2n + 1)! B 2n+1  x 2π  , where 0 ≤ x ≤ 2π for n = 1, 2, ; 0 < x < 2π for n = 0;andtheB n (x) are Bernoulli polynomials. . + 1)x] 2 cos x . 10. n  k=1 sin 2 (kx)= n 2 – cos[(n + 1)x]sin(nx) 2 sin x . 1118 FINITE SUMS AND INFINITE SERIES 11. n  k=1 cos 2 (kx)= n 2 + cos[(n + 1)x]sin(nx) 2 sin x . 12. n–1  k=1 k. numbers. 15. ∞  k=1 1 (2k – 1) 2n = (2 2n–1 – 1)π 2n 2(2n)! |B 2n |;theB 2n are Bernoulli numbers. 1120 FINITE SUMS AND INFINITE SERIES 16. ∞  k=1 1 k2 k =ln2. 17. ∞  k=0 (–1) k n 2k = n 2 n 2 + 1 . 18. ∞  k=0 1 k! =