Question

I've solved most! Just need help with 9, 10, 14.

Answer 1

7. Correct

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9. By the ratio test, the series will converge if

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{b^{n+1}(x-a)^{n+1}}{\ln(n+1)}}{\frac{b^n(x-a)^n}{\ln n}}\right|$

The limit reduces to

$\displaystyle|b(x-a)|\lim_{n\to\infty}\frac{\ln n}{\ln(n+1)}=b|x-a|$

where $|b|=b$ because $b>0$ is given. So the series converges when

$b|x-a|$

This means the radius of convergence is $\dfrac1b$.

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10. By the ratio test, the series converges if

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{2(n+1)}}{(n+1)(\ln(n+1))^2}}{\frac{x^{2n}}{n(\ln n)^2}}\right|$

The limit is

$\displaystyle|x^2|$

and so the radius of convergence is 1.

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11. Incorrect. By the root test, the series converges for

$\displaystyle\lim_{n\to\infty}\sqrt[n]{\left|\frac{(x-2)^n}{n^n}\right|}=\lim_{n\to\infty}\frac{|x-2|}n=0$

which means the series converges for all $x$, and so the interval of convergence is $(-\infty,\infty)$.

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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.