(a) Every power series converges somewhere. O True O False

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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For each of the questions below, indicate if the statement is true (T) or false (F).
(a) Every power series converges somewhere.
True
False
(b) If f(x) has a power series representation about a point x = a with radius of convergence R > 0, i.e.
00
f(x) = C(x – a)" for all x E (-R + a, a + R),
n=0
then a power series for f'(x) about the point x = a with the same radius of convergence R can be obtained by
differentiating the power series for f(x) term-by-term, i.e.
00
f'(x) = > nc, (x – a)"-1 for all x E (-R + a, a + R).
%3D
n=0
True
False
Transcribed Image Text:For each of the questions below, indicate if the statement is true (T) or false (F). (a) Every power series converges somewhere. True False (b) If f(x) has a power series representation about a point x = a with radius of convergence R > 0, i.e. 00 f(x) = C(x – a)" for all x E (-R + a, a + R), n=0 then a power series for f'(x) about the point x = a with the same radius of convergence R can be obtained by differentiating the power series for f(x) term-by-term, i.e. 00 f'(x) = > nc, (x – a)"-1 for all x E (-R + a, a + R). %3D n=0 True False
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