Hello,
How can I show if these series are convergent or divergent?
∑k = 1∞ cos((k^(3))/(3^(k))
∑k = 1∞ ((-1)^(k))((sin(1/k))/(k))
Appriciate any guidance and a detail explanation.
Thank you
Hmm...
I thought the first one diverges...
Doesn't $\displaystyle \lim_{k \to \infty} \frac{k^3}{3^k} = 0$ which would mean the sum just becomes 1+1+1+1+1+... (or near enough 1)...
I just checked in Maple and you have...
$\displaystyle \sum_{k=1}^{n} \cos(\tfrac{3^k}{k^3})$ tends to n-1 (or n-1.3 or something like that...) for n large
The first one is pretty easy.
You've got:
$\displaystyle
\lim_{k \to \infty} cos(\frac{k^3}{3^k}) =cos( \lim_{k \to \infty} \frac{k^3}{3^k})=cos0=1 \neq 0$
and it is divergent. If it was 0 the test would've been inconclusive. (limit test)
For the second one:
you can see that $\displaystyle \lim_{x->\infty}\frac{sin(\frac{1}{x})}{x}=0$
and it is decreasing (by drawing the graphic of the function $\displaystyle \frac{sin(\frac{1}{x})}{x}$.
We can conclude that it is alternating series which converge.
Regards.