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Does the series $\sum \limits _{n=0}^{\infty} \cos(n\pi)$ converge or diverge?

On substituting values I get alternate $1$ and $-1$.

So taking sum of infinite GP, I get $\dfrac 1 2$. So it looks like it converges.

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    $\begingroup$ Try to write out some values of $a_n$. What do you see? $\endgroup$ Nov 26, 2015 at 18:52
  • $\begingroup$ @NigelOvermars I get altternate 1 and -1. $\endgroup$
    – user293702
    Nov 26, 2015 at 18:53
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    $\begingroup$ @user293702 So what does that tell you? $\endgroup$ Nov 26, 2015 at 18:56
  • $\begingroup$ @NigelOvermars I edited my question $\endgroup$
    – user293702
    Nov 26, 2015 at 18:58
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    $\begingroup$ @user293702 Are you talking about the sequence $\{a_n\}$ or the sum $S_n = a_1 + ... + a_n$? $\endgroup$ Nov 26, 2015 at 19:01

1 Answer 1

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The sequence $\{a_n\}$ defined by $a_n = \cos(n \pi)$ diverges, since we have $a_{2n} = 1$ and $a_{2n+1} = -1$ for every $n \in \mathbb{N} \cup \{ 0 \}$. Hence two different subsequences have different limits, which implies divergence .

The series (sum) $S_n = a_1 + ... + a_n$ also diverges, since we have, following the same line of reasoning, $S_{2n} = 0$ and $S_{2n+1}=-1$. Hence the series also diverges.

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  • $\begingroup$ What about the sum of an infinite GP which turns out to be 1/2 $\endgroup$
    – user293702
    Nov 26, 2015 at 19:12
  • $\begingroup$ You should check the conditions for applying the formula of a geometric progression. And look at this wikipedia page: en.wikipedia.org/wiki/Grandi%27s_series $\endgroup$ Nov 26, 2015 at 19:16
  • $\begingroup$ Gotcha. Thanks a lot. $\endgroup$
    – user293702
    Nov 26, 2015 at 19:18

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