Consider the following power series n23" Σ - 2)". n=0 (a) Show that the power series converges for all ar E R. (It might be useful to recall that lim (:) - n +1 n00 n23n (x – 2)", defined for all r E R. e so that lim :) Now, set f(x) = >) =.. n +1 n=0 (b) Find f(k)(2), for all k 2 0. (c) Find a power series representation for f(k)(x), for all k 2 0, and all a E R. 2.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Consider the following power series
²3"
(x – 2)".
n
n=0
n
n + 1
(a) Show that the power series converges for all x E R. (It might be useful to recall that lim
(„)
n23"
(x - 2)", defined for all x E R.
e so that lim
...) Now, set f(x) =
||
|
n +1
n=0
(b) Find f(k) (2), for all k >0.
(c) Find a power series representation for f(k) (x), for all k > 0, and all x E R.
(d) Find a power series representation for an antiderivative of f(x).
2.
Transcribed Image Text:Consider the following power series ²3" (x – 2)". n n=0 n n + 1 (a) Show that the power series converges for all x E R. (It might be useful to recall that lim („) n23" (x - 2)", defined for all x E R. e so that lim ...) Now, set f(x) = || | n +1 n=0 (b) Find f(k) (2), for all k >0. (c) Find a power series representation for f(k) (x), for all k > 0, and all x E R. (d) Find a power series representation for an antiderivative of f(x). 2.
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