Example (21) prove the validity of the following series of functionswith in the specified regenst-Cos (2-1)2-120∞ (-1)h (Z1j²n-1,(2n)!Mo- 06/2-1/200

We have to prove cosz-1z-1=∑n=0∞-1n2n!z-12n-1, 0<z-1<∞

Answered: Example (21) prove the validity of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Discuss the conditions for which Fourier series is convergent.
Question #4 of the attachment. Thank you in advance
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1) Write an equation for determining the nth term of the series 20 + 10 + 5 + 2.5 + ... A) None of these are correct b) t(n) = 1/2n +20 c) t(n) = 1/2 (n−1)+20 d) t(n)=20 . (1/2)^n e) t(n) = 20 ⋅ 1/2^n-1
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(2n)! Find a Maclaurin series for f(x). (Use for 1:3.5. (2n - 3).) --. 2"n!(2n-1) f(x) = 1 + f(x) = x + 16 n = 2
(2n+ 1)x Please explan Determine whether the series sin- converges or diverg es. If it converges, find its sum. thish L and how it nelotes to the n-D0 (4r + 1)x Since sin =1 and sin =-1 and the angle may be incremente by any multiple of 2x, then sin 2 =1 and (4r+3)a sin =-1, (any r). How does this nelate to the answer. Examine the partial sums of the series. The expanded series is sin 3n 5x + sin + sin ,+ sin + sin+ .. ansver Notice that the angles in consecutive even terms (n is even) differ by 2x, and the angles in consecutive odd terms (n is odd) differ by 2x. Since sin 0 = sin (0+2mx), all the even terms are equal to 1, and all the odd terms are equal to -1. Evaluating each term, the series is 1+(-1)+1+(-1)+1+ The partial sums altermate between 1 and 0, tending to (2n + 1)a neither value. Therefore, lim sin- k00 n=0 does not exist and the series diverges. detail. Pelease explain n thanke gou.
Evaluate the infinite series by identifying it as the value of a derivative of a geometric series. n(n 1) 5n ∞ n=2 = Hint: Write it as f" 5 where f(x) = ∞ n=0 n x" 25
The function f is defined by the power series (-1)" x²n f(x) = Σ n=0 (2n+1)! for all real numbers x. Use the first and second derivative test by finding f'(x) and f"(x). Determine whether f has a local maximum, a local minimum, or neither at x=0. x4 1+1-1 5! = 1 −²+ 3! 7! +. + (-1)"x²n (2n+1)!

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Example (21) prove the validity of the following series of functions with in the specified regansi- Cos (2-1) 2-1 en-1 (-1) (2-1²-1, o<12-112²00 no (2n)! عم