# Determine $N$ of series $\sum_{n=1}^{N}\frac{n^n}{(2n+1)!}$ so that it differs from the actual sum by less than $\frac{1}{200}$

I can establish that: $$\frac{n^n}{(2n+1)!}=\frac{n^n}{(2n)!(2n+1)}\le\frac{n^n}{n!}$$ But $$\sum_{n=1}^{+\infty}\frac{n^n}{n!}$$ diverges (by the ratio test). And even if it converged I wouldn't have info on the N that I need.

I might say that: $$\frac{n^n}{(2n+1)!}=\frac{n^n}{(2n)!(2n+1)}\le\frac{n^n}{2^n}=(\frac{n}{2})^n$$ which diverges as well.

I should be able to use the comparison test but I can't see how.

Let $$a_n=\frac{n^n}{(2n+1)!}$$ then for $$N,n\geq 1$$, $$\frac{a_{n+1}}{a_n}=\frac{(1+1/n)^n}{2(2n+3)}< \frac{e}{2(2n+3)}<\frac{3}{10}\implies a_{N+n}< a_N \left(\frac{3}{10}\right)^{n}$$ Moreover let $$S$$ be the sum of the series then $$0 Can you take it from here?
• Thanks for your answer. I'm probably missing a simple piece but I don't get why $\frac{a_{n+1}}{a_n}<\frac{3}{10}\implies a_{N+n}< a_N \left(\frac{3}{10}\right)^{n}$. Could you please clarify it? – zcb Apr 16 at 13:18
• @zcb $a_{n+1}<3/10a_n$ implies that $a_{N+1}<3/10a_N$ and $a_{N+2}<3/10a_{N+1}$. Then $a_{N+2}<3/10a_{N+1}<(3/10)^2a_N$. The general case can be shown in a similar way. – Robert Z Apr 16 at 15:26
Note that\begin{align}\frac{\dfrac{(n+1)^{n+1}}{(2n+3)!}}{\dfrac{n^n}{(2n+1)!}}&=\frac{(n+1)^{n+1}}{n^n}\times\frac{(2n+1)!}{(2n+3)!}\\&=(n+1)\left(1+\frac1n\right)^n\times\frac1{(2n+2)(2n+3)}\\&=\frac12\left(1+\frac1n\right)^n\frac1{2n+3}\\&<\frac1{2n+3}\text{ since \left(1+\frac1n\right)^nSo,\begin{align}\left(\sum_{n=1}^\infty\frac{n^n}{(2n+1)!}\right)-\left(\sum_{n=1}^N\frac{n^n}{(2n+1)!}\right)&=\sum_{n=N+1}^\infty\frac{n^n}{(2n+1)!}\\&=\frac{(N+1)^{N+1}}{(2N+3)!}+\frac{(N+2)^{N+2}}{(2N+5)!}+\cdots\\&<\frac{(N+1)^{N+1}}{(2N+3)!}+\frac{(N+1)^{N+1}}{(2N+3)!}\times\frac12+\frac{(N+1)^{N+1}}{(2N+3)!}\times\frac1{2^2}+\cdots\\&=\frac{(N+1)^{N+1}}{(2N+3)!}\left(1+\frac12+\frac1{2^2}+\cdots\right)\\&=2\frac{(N+1)^{N+1}}{(2N+3)!}.\end{align}Since$$2\frac{4^4}{9!}=\frac1{2\,835}<\frac1{200},$$taking $$N=3$$ will be enough.
• Thanks for your answer. I can't understand why $\sum_{n=N+1}^\infty\frac{n^n}{(2n+1)!}<\frac{N^N}{(2N+1)!}\sum_{n=N+1}^\infty\frac1{2^n}$. Could you please clarify it? – zcb Apr 16 at 14:40
Hint: try some stronger lower bound for denominator. For example, $$(2n + 1)! = \prod\limits_{i=1}^{2n+1} i = \prod\limits_{i=1}^{n} i \cdot \prod\limits_{i=n+1}^{2n+1} i > 2^{n - 1} \cdot n^n$$.