Find the interval of convergence of (x^n)/(n +1)

In summary, the conversation discusses using the ratio test to find the interval of convergence and radius of convergence for a given function. The person is unsure of how to proceed and asks if it is safe to assume the limit equals 0 and the interval of convergence is from -∞ to ∞ with a radius of convergence of ∞. The expert advises against assuming and encourages the person to prove it, and also asks if they know what n! is.
  • #1
Painguy
120
0

Homework Statement


My main issue here is that if I use the ratio test I end up with lim (x(n!+1))/((n+1)!+1) n-> ∞

I don't know how to progress here. I believe that the limit will equal 0 and so it's interval of convergence is from -∞<x<∞ with a Radius of convergence of ∞. Is it safe for me to assume that?

Homework Equations





The Attempt at a Solution

 
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  • #2
It is never safe to assume anything. Prove it.

ehild
 
  • #3
ehild said:
It is never safe to assume anything. Prove it.

ehild

Well that's why I was asking. I'm not sure how to deal with the factorials here.
 
  • #4
Well, do you know what n! is?

ehild
 

FAQ: Find the interval of convergence of (x^n)/(n +1)

What is the interval of convergence for this series?

The interval of convergence for this series is -1 < x < 1.

How do you determine the interval of convergence?

The interval of convergence can be determined by using the ratio test, where if the limit of |an+1/an| as n approaches infinity is less than 1, the series will converge. The endpoints of the interval need to be checked separately for convergence.

Does the value of n affect the interval of convergence?

Yes, the value of n does affect the interval of convergence. The larger the value of n, the smaller the interval of convergence will be.

Can the interval of convergence be infinite?

Yes, it is possible for the interval of convergence to be infinite. This would occur when the limit in the ratio test approaches 0, indicating that the series converges for all real values of x.

What happens if the value of x is outside of the interval of convergence?

If the value of x is outside of the interval of convergence, the series will diverge and the ratio test will not be applicable. This means that the series does not have a sum for those values of x.

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