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

1. mpj4

$\sum_{k=1}^{\infty} \frac{k^{60}}{e^k} = \lim_{k\to\infty} \frac{(k+1)^{60}}{e^{k+1}}*\frac{e^k}{k^{60}} = \lim_{k\to\infty} \frac{(k+1)^{60}}{e^{k}e}*\frac{e^k}{k^{60}} = \lim_{k\to\infty} \frac{(k+1)^{60}}{e}*\frac{1}{k^{60}}$

2. idku

I can't see the code

3. mpj4

4. ganeshie8

It is very very wrong to say that the series equals that limit

5. mpj4

ah, I was just checking for convergence.

6. idku

well 1/e, because limit n->0 of { (k+1)/k }^60 is 1

7. idku

I mean k -> ∞. ops

8. idku

(if you had k instead of 60 in the exponent, then it would be 1/e^2)

9. idku

but wit 60, it is still conv based ratio test, since |r|<1

10. mpj4

ahh, that never occurred to me.

11. idku

what do you mean?

12. idku

the ^k case?

13. mpj4

that I could just combine k+1 and 1/k to form ((k+1)/k)^(60)

14. mpj4

Thanks! I kept trying to find a factor to cancel out k^60, forgot about that property. I will close this now.

15. idku

Well, if we do algebra with limit properties: $\large \lim_{k \rightarrow \infty} \frac{(k+1)^{60}e^{k}}{k^{60}e^{k+1}}$ $\large \lim_{k \rightarrow \infty} \frac{(k+1)^{60}}{k^{60}e^{1}}$ $(1/e) \times \left(\large \lim_{k \rightarrow \infty} \frac{(k+1)^{60}}{k^{60}} \right)$ $(1/e) \times \left(\large \lim_{k \rightarrow \infty} (\frac{k+1}{k})^{60} \right)$ $(1/e) \times \left(\large \lim_{k \rightarrow \infty} \frac{k+1}{k} \right)^{60}$

16. idku

(1/e) times 1^(60) = 1/e

17. mpj4

yep yep. well thanks a lot, I can go to sleep now.

18. idku

lol, good night

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