How to test the convergence of
$$\sum_{n=0}^{\infty} \frac{nx^n}{n^3+x^{2n}}$$
I tried $\lim |\dfrac{a_{n+1}}{a_n}|$ approach but couldn't end anywhere.
Any hints!
How can I test this the convergence of $\sum \frac{nx^n}{n^3+x^{2n}}$.
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real-analysis
sequences-and-series
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0Pointwise or uniform convergence? Is $x\in \mathbb R?$ – 2017-02-19
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0Hint: for $|x| < 1$ the $x^{2n}$-term is negligible in the limit. For $|x| >1$ the $n^3$-term is negligible. I think $|x| = 1$ needs to be treated seperately but should be the easiest of the three. Find converging series of bigger and of smaller terms than the one you've got to sandwich it in. (It chould converge for all real or even complex x). – 2017-02-19
1 Answers
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HINT:
Note that $n^3+x^n>x^{2n}$ and hence
$$\left|\frac{nx^n}{n^3+x^{2n}}\right|\le n|x|^{-n}$$
And for $|x|\le 1$, $n^3+x^n>n^3-1$ and hence
$$\left|\frac{nx^n}{n^3+x^{2n}}\right|\le \frac{n}{n^3-1}$$