Question

power serie

Answer 1

\[\sum_{n=1}^{\infty} (-1)^{n+1}*\frac{ (x-1)^n }{ n }\]

convergence interval

a. (-1,1)
b.[0,2)
c. (0,2]
d.[0,2]
e.[-1,1]

Answer 2

hmm you are expanding around 1, so disregard answer a and e

Answer 3

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 \(x=2\)

Answer 4

do you know how to do that?

Answer 5

no, i don't

Answer 6

lets replace \(x\) by 2

Answer 7

the terms will look like \[(-1)^{n+1}\frac{(2-1)^n}{n}=(-1)^{n+1}\frac{1}{n}\]

Answer 8

this is an alternating series, whose terms go to zero, and so when you sum it, it will converge

Answer 9

now we repeat the process with \(x=0\) but before we do, is what i wrote above clear?