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Σ |(-1)"72n+1 (2n + 1)!22n+1° 2. Find the sum of the series n=1

Σ |(-1)"72n+1 (2n + 1)!22n+1° 2. Find the sum of the series n=1

Advanced Engineering Mathematics
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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Number 2 with neat handwriting please.

f (x) Maclaurin series Converges to ƒ (x) for All a = 1+x + n! +... 4! 2! 3! n=0 Σ (-1)"x²n+1 (2n + 1)! sin x 75 All æ +... 7! = T - 3! 5! n=0 Cos a 으 (-1)"z2n All x Σ (2n)! = 1- 2! +.. 6! 4! n=0 1 00 |리 < 1 > x" = 1+ x + x² + x* +x* +... 1- x n=0 1 00 E(-1)"x" = 1– æ+ a² – a* + æ* I> |리| 1+x n=0 In (1+ æ) (-1)"-'z x² |x| <1 and æ = 1 +... 4 n 2 3 n=1 (-1)"x2n+1 2n +1 tan |æ| <1 +... n=0 a (а — 1) (а E() (1+ æ)“ a (a – 1) :- 2) |æ| < 1 x" = 1+ ax + +... 2! 3! n=0 1. Find the Maclaurin series of the following functions (write your answer in summation notation), and calculate f(4(0). (a) f(x) = cos(x²). arctan x (b) f(x) 1 (c) f(x) V1+ 2x2 f(2) = / sin(x²)dx. (d) (-1)"72n+1 (2n + 1)!22n+1° 2. Find the sum of the series n=1
Transcribed Image Text:f (x) Maclaurin series Converges to ƒ (x) for All a = 1+x + n! +... 4! 2! 3! n=0 Σ (-1)"x²n+1 (2n + 1)! sin x 75 All æ +... 7! = T - 3! 5! n=0 Cos a 으 (-1)"z2n All x Σ (2n)! = 1- 2! +.. 6! 4! n=0 1 00 |리 < 1 > x" = 1+ x + x² + x* +x* +... 1- x n=0 1 00 E(-1)"x" = 1– æ+ a² – a* + æ* I> |리| 1+x n=0 In (1+ æ) (-1)"-'z x² |x| <1 and æ = 1 +... 4 n 2 3 n=1 (-1)"x2n+1 2n +1 tan |æ| <1 +... n=0 a (а — 1) (а E() (1+ æ)“ a (a – 1) :- 2) |æ| < 1 x" = 1+ ax + +... 2! 3! n=0 1. Find the Maclaurin series of the following functions (write your answer in summation notation), and calculate f(4(0). (a) f(x) = cos(x²). arctan x (b) f(x) 1 (c) f(x) V1+ 2x2 f(2) = / sin(x²)dx. (d) (-1)"72n+1 (2n + 1)!22n+1° 2. Find the sum of the series n=1
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