Answer to Question #161768 in Calculus for Vishal

Question #161768

Show that ∞n=1 (x^n)/(n^n) converges for all x ∈ R


1
Expert's answer
2021-02-12T18:06:46-0500

Let us show that n=1xnnn\sum_{n=1}^{\infty} \frac{x^n}{n^n} converges for all xR.x \in\mathbb R. Let us use Cauchy's root convergence test for series:

L=limnxnnnn=limnxn=xlimn1n=x0=0L=\lim\limits_{n\to\infty}\sqrt[n]{|\frac{x^n}{n^n}|}=\lim\limits_{n\to\infty}\frac{|x|}{n}=|x|\cdot\lim\limits_{n\to\infty}\frac{1}{n}=|x|\cdot 0=0


Since L<1L<1 for xR,x\in\mathbb R, we conclude that n=1xnnn\sum_{n=1}^{\infty} \frac{x^n}{n^n} converges for all xR.x \in\mathbb R.




Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment