I need to see if this series $\sum_{n=1}^\infty\limits \left(\frac {1+\cos(n)}3 \right)^n$ either converges or diverges. I was thinking that because the inside terms are going to fluctuate between $(0,\frac 23)$, the inside is never negative, so it's going to diverge because a sum of positive numbers raised to a power are strictly increasing? Is my logic correct here and/or if there is a theorem that strengthens my argument, it would be appreciated.
2 Answers
$\begingroup$
$\endgroup$
3
Hint: $$ \sum_{n=1}^\infty \left|\frac{1+\cos(n)}3 \right|^n \leq\sum_{n=1}^\infty \left(\frac23\right)^n<\infty, $$ hence the series converges absolutely, hence converges.
-
2
-
$\begingroup$ @Goobys Too detailed? But, as we may deduce fro the question, OP will have enough work with the remaining details. $\endgroup$– Kola B.Nov 4, 2015 at 1:21
-
1$\begingroup$ What I was referring to was the fact that you said "Hint", yet still gave the entirety of the solution in your answer. A hint is not supposed to give the whole answer. $\endgroup$– RickNov 4, 2015 at 1:45
$\begingroup$
$\endgroup$
The series converges. As you noted $1 + \cos(n) \leq 2$ so we have:
$\Sigma_{n=1}^{\infty} \frac{1+\cos(n)}{3} \leq \Sigma_{n=1}^{\infty} (2/3)^n = 2$ by the geometric series forumla.