<s>[INST] Determine if the following series is convergent. Determine if the following series is convergent:
$$\sum_n^{\infty} \frac{1}{n^2 + \cos \pi n}$$.
 [/INST] $\dfrac{1}{n^2-1}\geqslant\dfrac{1}{n^2+\cos\pi n}$  
So, if $\sum\dfrac{1}{n^2-1}$ then $\sum\dfrac{1}{n^2+\cos\pi n}$ converges.  
Now, by Cauchy Condensations Test, if $\sum\dfrac{2^n}{2^{2n}-1}$ converges, the sum converges.  
$\dfrac{2^n}{2^{2n}-1}=\dfrac{2^n}{(2^n+1)(2^n-1)}<\dfrac{1}{2^n-1}$  
We know that $\sum\dfrac{1}{2^n-1}$ converges.  
Hence, converges.
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