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June 2nd, 2017, 12:32 AM   #1
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Which of the following series converges to e

Which of the following series converges to $e$

A) $\displaystyle (1+ \frac{1}{2n})^n $
B) $\displaystyle (2+ \frac{1}{2!}+\frac{1}{3!}+...+\frac{1}{n!}) $
C) $\displaystyle (1+ \frac{1}{n})^n $
D) $\displaystyle ( \frac{2n+1}{2n-2})^n $

My answers are options A and C.
Please correct me if I'm wrong
Thank you
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June 2nd, 2017, 01:20 AM   #2
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What does (B) converge to? The other choices aren't series, so what did you do when considering them?
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June 3rd, 2017, 06:14 PM   #3
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Sequence C converges to e.
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June 17th, 2017, 04:19 AM   #4
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"Sequence" is the right word, not "series".

In (A), if we let i= 2n then n= i/2 so $\displaystyle \left(1+ \frac{1}{2n}\right)^n$ becomes $\displaystyle \left(1+ \frac{1}{i}\right)^{i/2}= \left[\left(1+ \frac{1}{i}\right)^i\right]^{1/2}$.

Last edited by skipjack; June 17th, 2017 at 05:11 AM.
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