Show that ∞n=1 (x^n)/(n^n) converges for all x ∈ R
Let us show that ∑n=1∞xnnn\sum_{n=1}^{\infty} \frac{x^n}{n^n}∑n=1∞nnxn converges for all x∈R.x \in\mathbb R.x∈R. Let us use Cauchy's root convergence test for series:
L=limn→∞∣xnnn∣n=limn→∞∣x∣n=∣x∣⋅limn→∞1n=∣x∣⋅0=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=0L=n→∞limn∣nnxn∣=n→∞limn∣x∣=∣x∣⋅n→∞limn1=∣x∣⋅0=0
Since L<1L<1L<1 for x∈R,x\in\mathbb R,x∈R, we conclude that ∑n=1∞xnnn\sum_{n=1}^{\infty} \frac{x^n}{n^n}∑n=1∞nnxn converges for all x∈R.x \in\mathbb R.x∈R.
Need a fast expert's response?
and get a quick answer at the best price
for any assignment or question with DETAILED EXPLANATIONS!
Comments
Leave a comment