I have the following series:
$\sum_n{a_n}$, where $a_1 = 1$, $a_{n+1} = a_n \,\frac{2 + \cos n}{\sqrt n}$. Does it converge or diverge?
Edit: formatted.
I have the following series:
$\sum_n{a_n}$, where $a_1 = 1$, $a_{n+1} = a_n \,\frac{2 + \cos n}{\sqrt n}$. Does it converge or diverge?
Edit: formatted.
Use the Ratio Test. You can read about it here.
Consider the sequence $a_{n+1}=a_n\frac{3}{\sqrt{n}}$. See whether it converges or not. Now see if this converges, why should the original sequence converge.
Continuing with Alex's answer: apply now Dirichlet's test*, with the monotone descending sequence $\,\displaystyle{\left\{\frac{1}{\sqrt n}\right\}}\,$ and the bounded sequence $\,\displaystyle{\left\{\sum_{k=1}^n\cos n\right\}}\,$
Alternatively, use direct comparison with a geometric series, noting that all terms are positive. The first 15 terms sum to whatever they sum to. After that, with $n\ge16$, then $a_{n+1}=a_n\frac{2+\cos(n)}{\sqrt{n}}\le\frac{3}{4}a_n$. Inductively, $a_n\le\left(\frac34\right)^{n-16}a_{16}$ for $n\geq16$. So the series sums to at most $\sum_{n=1}^{15}a_n+a_{16}\sum_{n=16}^\infty\left(\frac34\right)^{n-16}$ or rather $\sum_{n=1}^{15}a_n+4a_{16}$