How do I solve this step by step?
1 answer:
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7. Correct
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9. By the ratio test, the series will converge if
The limit reduces to
where
because 
is given. So the series converges when
This means the radius of convergence is
.
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10. By the ratio test, the series converges if
The limit is
and so the radius of convergence is 1.
- - -
11. Incorrect. By the root test, the series converges for
![\displaystyle\lim_{n\to\infty}\sqrt[n]{\left|\frac{(x-2)^n}{n^n}\right|}=\lim_{n\to\infty}\frac{|x-2|}n=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Csqrt%5Bn%5D%7B%5Cleft%7C%5Cfrac%7B%28x-2%29%5En%7D%7Bn%5En%7D%5Cright%7C%7D%3D%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7B%7Cx-2%7C%7Dn%3D0%3C1)
which means the series converges for all
, and so the interval of convergence is 
.
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For 14 and 16, it'll probably be too late to edit this post by the time you see this. You can try posting the remaining problems in a new question.
- - -
9. By the ratio test, the series will converge if
The limit reduces to
where
This means the radius of convergence is
- - -
10. By the ratio test, the series converges if
The limit is
and so the radius of convergence is 1.
- - -
11. Incorrect. By the root test, the series converges for
which means the series converges for all
- - -
For 14 and 16, it'll probably be too late to edit this post by the time you see this. You can try posting the remaining problems in a new question.