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Hi! I am working on some online calc2 homework problems on power series and I am completely confused on how to solve these types of questions. I really do not know how to begin to tackle this problem, so if someone has a free minute to help me out I would really appreciate it!

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Let $$u_n(x)=n(x+4)^n$$ then by the ratio test we have $$\lim_{n\to\infty}\left|\frac{u_{n+1}(x)}{u_n(x)}\right|=|x+4|<1\iff x\in (-5,-3)$$ hence the interval of convergence is $(-5,-3)$. (We verify that the series isn't convergent on $x=-5$ and $x=-3$.)

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A power series $\sum a_n(x-a)^n$ will converge in an interval with center $a$ and radius $$R=\left(\limsup\frac{|a_{n+1}|}{|a_n|}\right)^{-1}$$.

In your case $a=-4$, and $a_n=n$.

Then the boundary points $-4+R$ and $-4-R$ have to be studied directly in the series. You will put $x=-4+R$ and study if the series converges or not. Similarly for $x=-4-R$.

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