Question

SERIES, CONVERGENCE
http://i1084.photobucket.com/albums/j409/QRAWarrior/MATA36-2.png

I know this theorem:
Given a geometric series (G.S.), if |r| < 1 the series CONV (with sum a/(1-r)), if |r| >= 1 then the series DIV.

But, it seems to not be "Working"...

Answer 1

\[
\sum_{n=0}^{\infty}\frac{(\cos(x))^n}{7^n}\\
\sum_{n=0}^{\infty}\left(\frac{\cos(x)}{7}\right)^n\\
\left|\frac{\cos(x)}{7}\right|\leq\frac{1}{7}<1
\]

Answer 2

Alright ,but then what would be the the values for x for which this converges?

Answer 3

What I'm trying to tell you is that it doesn't matter, as it converges for any \(x\in\mathbb{R}\).

Answer 4

I do not think that is true, because you just showed that it must be less than 1.

Answer 5

So would not it be from (-infinity, 1)?

Answer 6

Notice that\[
\sum_{n=0}^{\infty}\frac{(\cos(x))^n}{7^n}=-\frac{7}{\cos(x)-7}
.\]This converges only when \(|\cos(x)|<7\), but that's always true for any \(x\).

Answer 7

TRUE!