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.
$\begingroup$
$\endgroup$
10
-
4$\begingroup$ Try to write out some values of $a_n$. What do you see? $\endgroup$– Nigel OvermarsNov 26, 2015 at 18:52
-
$\begingroup$ @NigelOvermars I get altternate 1 and -1. $\endgroup$– user293702Nov 26, 2015 at 18:53
-
1$\begingroup$ @user293702 So what does that tell you? $\endgroup$– Nigel OvermarsNov 26, 2015 at 18:56
-
$\begingroup$ @NigelOvermars I edited my question $\endgroup$– user293702Nov 26, 2015 at 18:58
-
1$\begingroup$ @user293702 Are you talking about the sequence $\{a_n\}$ or the sum $S_n = a_1 + ... + a_n$? $\endgroup$– Nigel OvermarsNov 26, 2015 at 19:01
|
Show 5 more comments
1 Answer
$\begingroup$
$\endgroup$
3
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.
answered Nov 26, 2015 at 19:10
-
$\begingroup$ What about the sum of an infinite GP which turns out to be 1/2 $\endgroup$ 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
-