## lopus Group Title power serie one year ago one year ago

1. lopus Group Title

$\sum_{n=1}^{\infty} (-1)^{n+1}*\frac{ (x-1)^n }{ n }$ convergence interval a. (-1,1) b.[0,2) c. (0,2] d.[0,2] e.[-1,1]

2. satellite73 Group Title

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

3. satellite73 Group Title

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 $$x=2$$

4. satellite73 Group Title

do you know how to do that?

5. lopus Group Title

no, i don't

6. satellite73 Group Title

lets replace $$x$$ by 2

7. satellite73 Group Title

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

8. satellite73 Group Title

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

9. satellite73 Group Title

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