For what values of x does this series converge

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The series defined by (ln x)^n, where n ranges from 1 to infinity, converges when the absolute value of ln x is less than 1. This condition can be derived from the ratio test, which indicates that the series converges if |ln x| < 1. Therefore, the series converges for values of x in the interval (e^(-1), e). The discussion emphasizes the importance of understanding the behavior of logarithmic functions in relation to series convergence.

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Homework Statement


Series is:

(ln x)^n, n goes from 1 to infinity

Homework Equations




The Attempt at a Solution


For other problems I've seen the ratio test used to find the radius of convergence, but I don't think this can work here. What other things can I do to find where this converges?
 
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For what values of x does the series:

\sum_{n=1}^{\infty}x^n

converge?
 
Last edited:
Gauss177 said:

Homework Statement


Series is:

(ln x)^n, n goes from 1 to infinity

Homework Equations




The Attempt at a Solution


For other problems I've seen the ratio test used to find the radius of convergence, but I don't think this can work here. What other things can I do to find where this converges?

Why wouldn't the ratio test work here? The ratio test, remember, works for every infinite series, not just power series.
\frac{(ln x)^n}{(ln x)^{n+1}}= \frac{1}{ln x}
In order that the series converge, that fraction must be less than 1.

(And, of course, that gives the same answer as mattmns' solution.)
 
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