## lopus 3 years ago power serie

1. lopus

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

2. anonymous

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

3. anonymous

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. anonymous

do you know how to do that?

5. lopus

no, i don't

6. anonymous

lets replace $$x$$ by 2

7. anonymous

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

8. anonymous

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

9. anonymous

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