Tricky series radius of convergence question (analysis course)

In summary, the conversation discusses finding the radius of convergence for the series 1/(n^n)*x^(2^n) using various tests, such as the ratio test and nth root test. The speaker suggests using a combination of l'Hopital or a ratio test with a comparison test to show that the series diverges for values of x greater than 1. They also mention that the case for |x|<1 is easier to prove.
  • #1
pcvt
18
0

Homework Statement


Find the radius of convergence of sum from 1 to n of

1/(n^n) * x^(2^n)


Homework Equations





The Attempt at a Solution


Clearly ratio test isn't going to work straight away. I'm not sure how to deal with the 2^n exponent
 
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  • #2
What about the nth root test?
 
  • #3
From there I get down to finding for what values the limsup of 1/n x^(2^n/n) is less than one for

I believe its |x|<=1 but I'm not sure how to analytically prove this
 
  • #4
pcvt said:
From there I get down to finding for what values the limsup of 1/n x^(2^n/n) is less than one for

I believe its |x|<=1 but I'm not sure how to analytically prove this

I don't think |x|=1 works. But the case |x|<1 is easy. And |x|>1 is not much harder, you just want to show it diverges somehow. I'd say the easiest way is to combine l'Hopital or a ratio test with a comparison test. For example, can you show 2^n/n>n for large enough values of n?
 
Last edited:

1. What is a series and what is a radius of convergence?

A series is an infinite sequence of numbers, where each term is added to the previous term to create a sum. The radius of convergence is a value that determines the interval in which the series will converge, or approach a finite limit.

2. How is the radius of convergence determined for a tricky series?

The radius of convergence for a tricky series is determined by using the ratio test or the root test. These tests analyze the behavior of the terms in the series to determine if it will converge or diverge.

3. Can the radius of convergence be negative?

No, the radius of convergence can only be positive or zero. Negative values do not make sense in the context of a series, as it represents the distance from the center of convergence.

4. What does it mean if the radius of convergence is infinite?

If the radius of convergence is infinite, it means that the series will converge for all values of the variable. This is also known as a power series, where the series can be represented by a polynomial function.

5. How can the radius of convergence be used to determine the convergence of a series?

The radius of convergence provides a range of values for which the series will converge. If the value of the variable falls within this range, the series will converge. If the value is outside of this range, the series will diverge.

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