Radius of Convergence for Series with (-1)^(n-1) and x^n Terms

[tracedinair]

Homework Statement

Find the radius of convergence of \sum n=1 to ∞ of [(-1)^(n-1) x^(n)/(n)] and give a formula for the value of the series at the right hand endpoint.

Homework Equations

The Attempt at a Solution

Not really sure how to start this. I know I'm supposed to use the root test or the ratio test and then probably Abel's Thm for series. but the n-1 in the series is throwing me off. I've never seen that before. Any help is appreciated.

 

[Dick]

Your series is just f(x)=x-x^2/2+x^3/3+... Just use the ratio test to find the radius of convergence. To find the actual value at the endpoint you could just ask yourself is you have seen that kind of series before. If it doesn't look familiar, try finding f'(x) and see if that series looks familiar.

 

[tracedinair]

Your series is just f(x)=x-x^2/2+x^3/3+... Just use the ratio test to find the radius of convergence. To find the actual value at the endpoint you could just ask yourself is you have seen that kind of series before. If it doesn't look familiar, try finding f'(x) and see if that series looks familiar.

So I took f'(x) = 1 - x + x^2 - x^3 + ... so, the endpoint is going to look like x^(n+1)? I need a formula for the value at the right hand endpoint..would that work?

 

[Dick]

No, you have to recognize the series, my point to looking at f'(x) was that it's geometric, it's 1/(1+x). if that's f'(x), what's f(x)?

 

[tracedinair]

If f'(x) = 1/(1+x) then f(x) = ln(x+1)?

 

[Dick]

ln(x+1)+C, you mean. You have to fix the constant C by finding f(0). But yes, that's the idea.

 

[tracedinair]

Alright, thank you.

 

[HallsofIvy]

Homework Statement

Find the radius of convergence of \sum n=1 to ∞ of [(-1)^(n-1) x^(n)/(n)] and give a formula for the value of the series at the right hand endpoint.

Homework Equations

The Attempt at a Solution

Not really sure how to start this. I know I'm supposed to use the root test or the ratio test and then probably Abel's Thm for series. but the n-1 in the series is throwing me off. I've never seen that before. Any help is appreciated.

The ratio test applies to a positive series and a power series converges absolutely inside its "radius of convergence".

Take the absolute value (drop the "(-1)^{n-1}") and then use the ratio test.