TITLE: What is the nature of the series $\sum \frac{5^n - 3^n}{3^n + n^4 } $
QUESTION [3 upvotes]: What is the nature of the series with the general term $u_n = \frac{5^n - 3^n}{3^n + n^4 } $ ?


I have tried to find an inequality from which I can derive the convergence or non convergence of the serie but I could not progress much beyond: 
$ u_n \geq \frac{5^n}{3^n + n^4 } \ge \frac{3^n}{3^n + n^4 } $ 
And: $\lim  \frac{3^n}{3^n + n^4 }  = 1$
which is not helpful. 
I tried also to write $(u_n)$ in the exponential form but I could not proceed either.
What convergence criteria will be used in this case? 
Thank you.

REPLY [3 votes]: Hint: We get
$$\lim_{n->\infty}\frac{5^n-3^n}{3^n+n^4}=\infty$$
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