# Using Ratio test to determine convergence

In this series I need to use the Ratio test to determine whether the series diverges or converges and I'm not sure about my calculations $$\sum_{n=1}^\infty \left( \frac{\pi}{n}\right)^n * n!$$ plug in n+1 * 1/n

$$\left( \frac{\pi}{n+1}\right)^{n+1} (n+1)! * \left( \frac{n}{\pi}\right)^{n} \left( \frac{1}{n!}\right)$$

$$\left( \frac{\pi}{n+1}\right)^{n+1} (n+1)* \left( \frac{n}{\pi}\right)^{n}$$ $$\left( \frac{\pi}{n+1}\right)^{n} \pi* \left( \frac{n}{\pi}\right)^{n}$$ $$\left( \frac{n}{n+1}\right)^{n} \pi$$ $$\left( \frac{1}{2}\right)^{n} \pi = 0$$ which would be 0 so it is convergent ??

• $(n/n+1)^n \ne (1/2)^n$. – Tito Eliatron Oct 29 '20 at 9:49

$$\left(\frac{n}{n+1} \right)^n =\frac{1}{\left(\frac{n+1}{n}\right)^n} =\frac{1}{\left(1+\frac{1}{n}\right)^n}\to \frac{1}{e}$$

So the limit of the ratio test should be $$\dfrac{\pi}{e}>1$$, so your series is divergent.

• ohh Thank you ! – GregoryStory16 Oct 29 '20 at 9:54