- #1
Sheneron
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Homework Statement
Show that the series converges. Then compute an estimate of the limit that is guaranteed to be in error by no more than 0.005
[tex]\sum_{k=5}^{\infty} (-1)^k \frac{k^{10}}{10^k}[/tex]
The Attempt at a Solution
This is obviously an alternating series and I know that
[tex]C_{k} = \frac{k^{10}}{10^k}[/tex]
and I know that Cn+1 is greater than the absolute value of S-Sn. So I can set up to be something like
[tex]\frac{(n+1)^{10}}{10^{n+1}} <= 0.005[/tex]
the part I can't figure out is how to solve that for n. Is there a way to simplify that fraction? How would I solve this for n? Thanks
Oh and I also couldn't figure out how to exactly show that the limit as k-> infinity of Ck goes to 0 without taking the derivative 10 times. So the whole problem I am having is with the fraction.