Use term-by-term differentiation or integration to find a power series centered at0 for:-877f(x)(1+ z*)?

Answered: Use term-by-term differentiation or…

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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Question

Use term-by-term differentiation or integration to find a power series centered at
0 for:
-877
f(x)
(1+ z*)?

Expert Solution

Step 1

$A p o w e r s e r i e s o f t h e g i v e n f u n c t i o n u \sin g t e r m b y t e r m d i f f e r e n t i a t i o n c e n t e r e d a t x = 0 i s g i v e n b y f (x) = \sum_{k = 0}^{\infty} \frac{f^{k} (0)}{k!} x^{k} s o, w e w i l l f i n d d e r i v a t i v e o f t h e g i v e n f u n c t i o n a n d i t s v a l u e a t t h e g i v e n p o i n t x = 0 F i r s t d e r i v a t i v e i s - f (x) = \frac{- 8 x^{7}}{{(1 + x^{8})}^{2}} f' (x) = \frac{{(1 + x^{8})}^{2} (- 8) 7 x^{6} + 8 x^{7} \cdot 2 (1 + x^{8}) 8 x^{7}}{{(1 + x^{8})}^{4}} = \frac{8 (1 + x^{8}) [- 7 x^{6} (1 + x^{8}) + 16 x^{14}]}{{(1 + x^{8})}^{4}} = \frac{8 [- 7 x^{6} + 9 x^{14}]}{{(1 + x^{8})}^{3}} = \frac{8 x^{6} (9 x^{8} - 7)}{{(1 + x^{8})}^{3}} f' (0) = 0$

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