calculus.
Is $\sum_{n=2}^\infty \frac{n!}{(n-1)^n}$ convergent?
i tried ratio test and can't continue. I also tried the root test.
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I assume you mean you tried the ratio and root test? – user12345 May 10 '17 at 00:02
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yes i meant that thanks – Allen Zheng May 10 '17 at 00:06
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Perhaps you should show how to tried the ratio test. As I've shown below, it should work out just fine. Root test I think would actually work also assuming you know the limit $$\lim_{n\to\infty}\sqrt[n]{n!}$$ – user12345 May 10 '17 at 00:07
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i dont know that limit. what is it? – Allen Zheng May 10 '17 at 00:08
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Well, here's a link to that limit: https://math.stackexchange.com/questions/514388/the-nth-root-of-n?rq=1 however, my memory was mistaken on it and I don't think the root test works well for this example – user12345 May 10 '17 at 00:13
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Ratio test: $$\lim_{n\to\infty}\dfrac{(n+1)!}{n^{n+1}}\cdot\dfrac{(n-1)^n}{n!}=\lim_{n\to\infty}\dfrac{n+1}{n}\cdot\left(\dfrac{n-1}{n}\right)^n=\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^n=e^{-1}<1$$
user12345
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did you just assume that the 1+1/n would be 0 at infinity so you didn't put it in the last limit? – Allen Zheng May 10 '17 at 00:07
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No! You can not do that because the exponent is not a constant. This is a well-known type of limit. Here is a link for you to check out: https://math.stackexchange.com/questions/358830/about-lim-left1-frac-xn-rightn and especially: https://math.stackexchange.com/questions/370125/why-isnt-lim-limits-x-to-infty-left1-frac1x-rightx-equal-to-1?rq=1 – user12345 May 10 '17 at 00:10