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 ??

  • $\begingroup$ $(n/n+1)^n \ne (1/2)^n$. $\endgroup$ – 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.

  • 1
    $\begingroup$ ohh Thank you ! $\endgroup$ – GregoryStory16 Oct 29 '20 at 9:54

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