What is the radius of convergence for a series with logarithmic terms?

[AllRelative]

Homework Statement

This is from a complex analysis course:

Find radius of convergence of
$$\sum_{}^{} (log(n+1) - log (n)) z^n$$

Homework Equations

I usually use the root test or with the limit of $\frac {a_{n+1}}{a_n}$

The Attempt at a Solution

My first reaction is that this sum looks like a telescoping sum. But since the term $z_n$ is there, it doesn't cancel out.

Then I thought I could split the sum in two:
$\sum_{}^{} log(n+1)z^n - \sum log (n)z^n$

But I am not sure whereto go from there.Thanks for the help!

 

[Buzz Bloom]

Hi AR:

I am guessing that the following is something you know, but may need to refresh your memory.

https://en.wikipedia.org/wiki/Radius_of_convergence​

It may also help you if you put the coefficients cⁿ into the form

c_n = log(f(n)).​

That is, express c_n as a single log function of an expression involving n.

Hope this helps.

Regards,
Buzz

 

[scottdave]

As @Buzz Bloom said, remember how logarithms work. For example, how could you rewrite log(a) - log(b) into a single log?

 

[AllRelative]

As @Buzz Bloom said, remember how logarithms work. For example, how could you rewrite log(a) - log(b) into a single log?

Log(n+1)-log(n) = log(1 + 1/n) then the root test is easy. Thanks man