Radius of convergence log(a + x)

[Lengalicious]

Homework Statement

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

Homework Equations

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...

 

[ehild]

So after applying the Taylor series expansion about x=0 we get log(a) + SUM[(-1)^n x^n/(n a^n)]

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

 

[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)?

 

[ehild]

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