Exercise Determine the interval of convergence of the power series n=1 n (2x + 1)". O Explain why the following solution to this exercise is mathematically incorrect and/or incomplete, identifying at least three errors or significant omissions. For each error or omission, explain the mistake that the writer of the solution has made. (There may be more than three errors or omissions, but you need identify only three. These should not include incorrect statements or omissions that follow logically from earlier errors or omissions.) Solution (incorrect and/or incomplete!) We have Στ 1 (2x+1)" = Σ. (1) n 1 n=1 and so, taking an 2 = an+1 2 = an n+1 - n we have n 22 n = → 1 as n→ ∞. n+1 +1 So the radius of convergence of this power series is R = 1. It follows that the interval of convergence is (- ½) O Give a correct solution to the exercise, justifying your answer carefully.
Exercise Determine the interval of convergence of the power series n=1 n (2x + 1)". O Explain why the following solution to this exercise is mathematically incorrect and/or incomplete, identifying at least three errors or significant omissions. For each error or omission, explain the mistake that the writer of the solution has made. (There may be more than three errors or omissions, but you need identify only three. These should not include incorrect statements or omissions that follow logically from earlier errors or omissions.) Solution (incorrect and/or incomplete!) We have Στ 1 (2x+1)" = Σ. (1) n 1 n=1 and so, taking an 2 = an+1 2 = an n+1 - n we have n 22 n = → 1 as n→ ∞. n+1 +1 So the radius of convergence of this power series is R = 1. It follows that the interval of convergence is (- ½) O Give a correct solution to the exercise, justifying your answer carefully.
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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Transcribed Image Text:Exercise Determine the interval of convergence of the power series
n=1
n
(2x + 1)".
O Explain why the following solution to this exercise is mathematically
incorrect and/or incomplete, identifying at least three errors or
significant omissions. For each error or omission, explain the mistake
that the writer of the solution has made. (There may be more than
three errors or omissions, but you need identify only three. These
should not include incorrect statements or omissions that follow
logically from earlier errors or omissions.)
Solution (incorrect and/or incomplete!)
We have
Στ
1
(2x+1)"
=
Σ. (1)
n
1
n=1
and so, taking an
2
=
an+1
2
=
an
n+1
-
n
we have
n
22
n
=
→
1 as n→ ∞.
n+1
+1
So the radius of convergence of this power series is R = 1. It follows
that the interval of convergence is (- ½)
O Give a correct solution to the exercise, justifying your answer carefully.
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