<s>[INST] Determine whether this series converges: Determine whether this series converges:
$ \sum_{n=1}^{\infty}\frac{\sqrt n\cos(n^2)}{n^{4/3}+\cos(n^2) }$
I tried
$$-1\leq \cos(n^2) \leq 1   \Rightarrow \sum_{n=1}^{\infty}\frac{- \sqrt n}{n^{4/3}-1 } \leq \sum_{n=1}^{\infty}\frac{\cos(n^2)\cdot \sqrt n}{n^{4/3}+\cos(n^2) } \leq \sum_{n=1}^{\infty}\frac{ \sqrt n}{n^{4/3}-1 }$$
Applying limit comparison test, we get
$- \infty \leq \sum_{n=1}^{\infty}\frac{\sqrt n\cos(n^2)}{n^{4/3}+\cos(n^2) } \leq \infty$

Comparison test doesn't work. What test should I use?
 [/INST] Denote
\begin{align}
a_n = \frac{\sqrt{n}}{n^{4/3} +\cos(n^2)},  \quad b_n = \cos(n^2), \quad \text{ and } \quad B_N = \sum^N_{n=0} \cos(n^2).
\end{align}
Then the given series can be expressed as
\begin{align}
\sum^\infty_{n=0} a_nb_n.
\end{align}
Now, using Abel's summation formula (or the so called summation-by-parts formula), we have that
\begin{align}
S_N:=\sum^N_{n=0} a_nb_n =&\, a_N B_N -\sum^{N-1}_{n=0} B_n\left(a_{n+1}-a_n\right)\\
=&\, \frac{\sqrt{N}}{N^{4/3} +\cos(N^2)}\sum^N_{n=0}\cos(n^2)+\sum^{N-1}_{n=0} \left(\frac{\sqrt{n}}{n^{4/3} +\cos(n^2)}-\frac{\sqrt{n+1}}{(n+1)^{4/3} +\cos((n+1)^2)}\right)\sum^n_{k=0}\cos(k^2).
\end{align}
First, notice that
\begin{align}
\frac{\sqrt{n}}{n^{4/3} +\cos(n^2)}-\frac{\sqrt{n+1}}{(n+1)^{4/3} +\cos((n+1)^2)} =&\, \frac{1}{n^{4/3}}\left(\frac{\sqrt{n}}{1 +\frac{\cos(n^2)}{n^{4/3}}}-\frac{\sqrt{n+1}}{(1+\frac{1}{n})^{4/3} +\frac{\cos((n+1)^2)}{n^{4/3}}}\right)\\
\simeq&\, \frac{1}{n^{4/3}\left(\sqrt{n+1}+\sqrt{n}\right)}\left(1+\mathcal{O}(\frac{1}{n^\alpha})\right)
\end{align}
provided $n$ is sufficiently large.
If we show that
\begin{align}
|B_N|=\left|\sum^N_{n=0} \cos(n^2)\right| \le C\sqrt{N} \log N
\end{align}
where $C$ is some constant independent of $N$ then we have that
\begin{align}
|S_N-S_M| \le&\, C\frac{N \log N}{N^{4/3} +\cos(N^2)}+C\frac{M \log M}{M^{4/3} +\cos(M^2)}\\
& +C\sum^{N-1}_{n=M} \left|\frac{\sqrt{n}}{n^{4/3} +\cos(n^2)}-\frac{\sqrt{n+1}}{(n+1)^{4/3} +\cos((n+1)^2)}\right|\left|\sum^n_{k=0}\cos(k^2)\right|\\
\le &\, C\frac{N \log N}{N^{4/3} +\cos(N^2)}+C\frac{M \log M}{M^{4/3} +\cos(M^2)}\\
&\, + C \sum^{N-1}_{n=M} \frac{\sqrt{n}\log n}{n^{4/3}\left(\sqrt{n+1}+\sqrt{n}\right)}.
\end{align}
Fix $\varepsilon>0$. Hence if we choose $K=K(\varepsilon)$ sufficiently large, then for any $M, N>K$ we have that
\begin{align}
|S_N-S_M|<\varepsilon.
\end{align}
This shows that the series converges.
The proof of the bound on $B_N$ can be found here.
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