What is the nature of the series $\sum \frac{5^n - 3^n}{3^n + n^4 } $

Question

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.

Answer 1

If a series $\sum u_n$ converges, then $\displaystyle\lim_{n\to\infty} u_n = 0$.

But $$ \lim_{n\to\infty} \frac{5^n - 3^n}{3^n + n^4 } = +\infty $$ and so the series diverges.

Answer 2

Your first inequality, that is, $u_n\geqslant\frac{5^n}{3^n+n^4}$, is a good starting point. Now, use the fact that$$\lim_{n\to\infty}\frac{\frac{5^n}{3^n+n^4}}{\frac{5^n}{3^n}}=1.$$

Answer 3

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