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mpj4
 one year ago
Calculus: Convergence test (ratio test): How to simplify this further?
mpj4
 one year ago
Calculus: Convergence test (ratio test): How to simplify this further?

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mpj4
 one year ago
Best ResponseYou've already chosen the best response.1\[\sum_{k=1}^{\infty} \frac{k^{60}}{e^k} = \lim_{k\to\infty} \frac{(k+1)^{60}}{e^{k+1}}*\frac{e^k}{k^{60}} = \lim_{k\to\infty} \frac{(k+1)^{60}}{e^{k}e}*\frac{e^k}{k^{60}} = \lim_{k\to\infty} \frac{(k+1)^{60}}{e}*\frac{1}{k^{60}} \]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0It is very very wrong to say that the series equals that limit

mpj4
 one year ago
Best ResponseYou've already chosen the best response.1ah, I was just checking for convergence.

idku
 one year ago
Best ResponseYou've already chosen the best response.2well 1/e, because limit n>0 of { (k+1)/k }^60 is 1

idku
 one year ago
Best ResponseYou've already chosen the best response.2(if you had k instead of 60 in the exponent, then it would be 1/e^2)

idku
 one year ago
Best ResponseYou've already chosen the best response.2but wit 60, it is still conv based ratio test, since r<1

mpj4
 one year ago
Best ResponseYou've already chosen the best response.1ahh, that never occurred to me.

mpj4
 one year ago
Best ResponseYou've already chosen the best response.1that I could just combine k+1 and 1/k to form ((k+1)/k)^(60)

mpj4
 one year ago
Best ResponseYou've already chosen the best response.1Thanks! I kept trying to find a factor to cancel out k^60, forgot about that property. I will close this now.

idku
 one year ago
Best ResponseYou've already chosen the best response.2Well, if we do algebra with limit properties: \[\large \lim_{k \rightarrow \infty} \frac{(k+1)^{60}e^{k}}{k^{60}e^{k+1}}\] \[\large \lim_{k \rightarrow \infty} \frac{(k+1)^{60}}{k^{60}e^{1}}\] \[(1/e) \times \left(\large \lim_{k \rightarrow \infty} \frac{(k+1)^{60}}{k^{60}} \right)\] \[(1/e) \times \left(\large \lim_{k \rightarrow \infty} (\frac{k+1}{k})^{60} \right)\] \[(1/e) \times \left(\large \lim_{k \rightarrow \infty} \frac{k+1}{k} \right)^{60}\]

mpj4
 one year ago
Best ResponseYou've already chosen the best response.1yep yep. well thanks a lot, I can go to sleep now.
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