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Classify $$\sum_{n=1}^\infty \frac{ x^2+\cos^n x}{n^2}$$ as absolutely convergent, conditionally convergent or divergent.

I am not sure whether I should use integral test or comparison test.

Do you know what's the best test to classify this series?

Thanks in advance.

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1 Answer 1

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Note:

$$\left | \sum_{n=1}^\infty \frac{ x^2+\cos^n x}{n^2}\right | \le \sum_{n=1}^\infty \frac{|x^2+\cos^n{x}|}{n^2} \le \sum_{n=1}^\infty \frac{ x^2+1}{n^2} = (x^2+1) \frac{\pi^2}{6}$$

Therefore the sum is absolutely convergent for any finite value of $x$ by the comparison test. Thanks to @N.S. and @vadim123 for correcting and clarifying the reasoning here.

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  • $\begingroup$ I misread, my bad. $\endgroup$
    – Pedro
    Jun 11, 2013 at 2:15
  • $\begingroup$ how did you did pie^2/6? $\endgroup$
    – Risa
    Jun 11, 2013 at 2:19
  • $\begingroup$ $$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$ This is a well-known sum. $\endgroup$
    – Ron Gordon
    Jun 11, 2013 at 2:21
  • $\begingroup$ oh i see. for this kind of series, is it best to use comparison test? $\endgroup$
    – Risa
    Jun 11, 2013 at 2:23
  • $\begingroup$ @Risa: yes, that's correct. $\endgroup$
    – Ron Gordon
    Jun 11, 2013 at 2:24

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