# Radius of convergence log(a + x)

1. Apr 25, 2014

### Lengalicious

1. The problem statement, all variables and given/known data

determine the radius of convergence of the series expansion of log(a + x) around x = 0

2. Relevant equations

3. The attempt at a solution

So after applying the Taylor series expansion about x=0 we get log(a) + SUM[(-1)^n x^n/(n a^n)] I understand how to get the radius of convergence for log(1+x) for instance using the ratio test, but with 1 being replaced with what I believe to be an arbitrary constant I am now confused as to how I should tackle this, any help would be great, thanks in advance!

Would I employ the ratio test in the same way? I have a log in the numerator and denominator so not really sure...

2. Apr 25, 2014

### ehild

There is a minor error: $$log(a+x)=log(a)+\sum_1^{\infty}(-1)^{n+1} \frac{x^n}{n a^n}$$
Do the same as with other power series.
Where do you have log in the terms of the series, except the first one?

ehild

Last edited: Apr 25, 2014
3. Apr 25, 2014

### Lengalicious

Yeah so I only have the log in the first term, that's why I am confused, just making sure it is the same as if all terms are just a power series, would I use the ratio test in the same vein as with log(1+x)?

4. Apr 25, 2014

### ehild

Yes, use the ratio test. Do not worry about the first term.

ehild