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

In summary, the conversation discusses finding the radius of convergence for the sum $$\sum_{}^{} (log(n+1) - log (n)) z^n$$ from a complex analysis course. One suggestion is to use the root test and another is to rewrite the coefficients in the form of a single logarithm. The conversation ends with the student understanding how to rewrite the coefficients and use the root test to find the radius of convergence.
  • #1
AllRelative
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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!
 
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  • #2
Hi AR:

I am guessing that the following is something you know, but may need to refresh your memory.
It may also help you if you put the coefficients cn into the form
cn = log(f(n)).​
That is, express cn as a single log function of an expression involving n.

Hope this helps.

Regards,
Buzz
 
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  • #3
As @Buzz Bloom said, remember how logarithms work. For example, how could you rewrite log(a) - log(b) into a single log?
 
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  • #4
scottdave said:
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
 
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Related to What is the radius of convergence for a series with logarithmic terms?

What is the definition of radius of convergence?

The radius of convergence is a concept in mathematics that is used to determine the interval of values for which a power series will converge. It is the distance from the center of the series to the point where the series converges.

How do you find the radius of convergence?

To find the radius of convergence, you can use the Ratio Test or the Root Test. These tests involve taking the limit of the ratio or root of the terms in the series. If the limit is less than 1, the series will converge, and the radius of convergence can be determined from this limit.

What is the significance of the radius of convergence?

The radius of convergence is important because it tells us the interval of values for which a power series will converge. This information is crucial in determining the validity and accuracy of mathematical models and calculations.

Can the radius of convergence be negative?

No, the radius of convergence cannot be negative. It represents a distance and therefore must be a positive value. If the radius of convergence is negative, it means that the series does not converge for any value of the variable.

How does the radius of convergence affect the convergence of a series?

The radius of convergence is directly related to the convergence of a series. If the radius of convergence is larger, the series will converge for a larger range of values. If the radius of convergence is smaller, the series will converge for a smaller range of values. If the radius of convergence is zero, the series will not converge at all.

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