Finding the Radius of Convergence for a Series with Exponential Growth

In summary, The conversation discussed the radius of convergence of the series \sum_{n \geq 1} n^{\sqrt{n}}z^n and the attempts to determine it using the ratio test and the root test. The latter was not successful, but it was suggested to use logarithms and techniques such as l'Hopital's theorem to find the limit of the log. It was also mentioned that the limit of n^(1/sqrt(n)) appears to be 1.
  • #1
Mathmos6
81
0
Hi there - I'm trying to work out the radius of convergence of the series [itex] \sum_{n \geq 1} n^{\sqrt{n}}z^n [/itex] and I'm not really sure where to get going - I've tried using the ratio test and got (not very far) with [itex]lim_{n \to \infty} | \frac{n^{\sqrt{n}}}{(n+1)^{\sqrt{n+1}}}|[/itex], and with the root test, [itex]\left( {lim sup_{n \to \infty} n^{\frac{1}{\sqrt{n}}}}\right) ^{-1} [/itex], neither of which seem to help me =/

I have a strong feeling the latter converges to 1 but even if I'm right I'm not totally sure how to prove it, and I may well be wrong. What should my next move be?

Thanks a lot!

Mathmos6
 
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  • #2
Don't give up so fast on the root test. What IS the limit n->infinity n^(1/sqrt(n))?
 
  • #3
I'm guessing 1, but I'm not sure how to prove it?
 
  • #4
Mathmos6 said:
I'm guessing 1, but I'm not sure how to prove it?

Take the log to turn it into a quotient. Try to find the limit of the log. Now you can use things like l'Hopital's theorem.
 
  • #5
That's brilliant, thanks! :)
 

1. What is the radius of convergence?

The radius of convergence is a term used in mathematics, particularly in calculus and series, to describe the interval or region within which a given power series will converge. It is typically denoted by the letter R.

2. How is the radius of convergence calculated?

The radius of convergence can be calculated using the ratio test, which involves taking the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term in the series. If this limit is less than 1, then the series converges, and the radius of convergence is equal to the reciprocal of the limit. If the limit is greater than 1, the series diverges, and if the limit is equal to 1, further tests are needed to determine the convergence or divergence of the series.

3. What is the significance of the radius of convergence?

The radius of convergence is important because it tells us the interval or region for which the power series converges. This information is useful in determining the behavior of the series and its sum. It also helps us to identify any singular points or discontinuities of the function represented by the series.

4. Can the radius of convergence be negative?

No, the radius of convergence cannot be negative. It is always a positive number or zero.

5. How does the radius of convergence relate to the interval of convergence?

The interval of convergence is the set of all values of x for which the power series converges. The radius of convergence is the distance from the center of the interval to either endpoint. In other words, the interval of convergence is the open interval centered at the center of convergence with a length equal to twice the radius of convergence.

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