Find whether $\sum_{n=1}^\infty\frac{(2n+1)!}{4^n+\frac{1}{π^n}}$ is convergent or divergent

Question

In this exercise I have to find whether this series is convergent or divergent, and I do not really know what to do with denominator, that is why I would be grateful for any tips or solutions.

$$\sum_{n=1}^\infty\frac{(2n+1)!}{4^n+\frac{1}{π^n}}$$

Interesting. I believe that from this answer: https://math.stackexchange.com/questions/351815/do-factorials-really-grow-faster-than-exponential-functions, the numerator would grow faster than the denominator and the function would eventually diverge. — TheBestMagician, Mar 14 '21 at 17:44

score 3 · Answer 1 · answered Mar 14 '21 at 18:38

Big hint: Consider the general term:

its numerator is clearly $> \ n!$
its denominator is less than $ \ < 5^n$. (you will have to provide a rigorous proof)

Therefore the general term is $>\dfrac{n!}{5^n}$ which tends to $\infty$...

As a consequence, the given series is divergent.

Find whether $\sum_{n=1}^\infty\frac{(2n+1)!}{4^n+\frac{1}{π^n}}$ is convergent or divergent

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