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lopus
power serie
\[\sum_{n=1}^{\infty} (-1)^{n+1}*\frac{ (x-1)^n }{ n }\] convergence interval a. (-1,1) b.[0,2) c. (0,2] d.[0,2] e.[-1,1]
hmm you are expanding around 1, so disregard answer a and e
radius of convergence is 1, so it is one of the middle three your job is to check at the endpoints, and see if it converges at \(x=0\) and if it converges at \(x=2\)
do you know how to do that?
lets replace \(x\) by 2
the terms will look like \[(-1)^{n+1}\frac{(2-1)^n}{n}=(-1)^{n+1}\frac{1}{n}\]
this is an alternating series, whose terms go to zero, and so when you sum it, it will converge
now we repeat the process with \(x=0\) but before we do, is what i wrote above clear?