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Consider the series

$ \sum_{n=1}^\infty x^2+ n/n^2$

.

Pick out the true statements:

(a) The series converges for all real values of $x$.

(b) The series converges uniformly on $\mathbb{R}$.

(c) The series does not converge absolutely for any real value of $x$.

stuck on this problem totally. please help me to solve this problem.thanks .

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Hint: $$\sum_{n=1}^\infty\frac{1}{n}=\sum_{n=1}^\infty\frac{n}{n^2}\leq\sum_{n=1}^\infty x^2+\frac{n}{n^2}.$$

If you inadvertently meant $(x^2+n)/n^2$, then $$\sum_{n=1}^\infty\frac{1}{n}=\sum_{n=1}^\infty\frac{n}{n^2}\leq\sum_{n=1}^\infty\frac{x^2+n}{n^2}.$$

Only (c) applies. because $\sum_{n\ge 1} {1/n}$ diverges..

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