Power series and finding radius of convergence

In summary, the problem is asking to find the radius of convergence and interval of convergence of the series \sum_{n=0}^\infty \frac{x^n}{n!}. The ratio test is used to determine the convergence of the series and the limit is taken as n approaches infinity. The radius of convergence is found to be infinite, indicating that the series converges for all values of x, and the interval of convergence is also infinite.
  • #1
cowmoo32
122
0

Homework Statement


"Find the radius of convergence and interval of convergence of the series"
[tex]\sum_{n=0}^\infty \frac{x^n}{n!}[/tex]

Homework Equations



Ratio Test

The Attempt at a Solution



[tex]\lim{\substack{n\rightarrow \infty}} |x/n+1|[/tex]
(I can't seem to get the |x/n+1| to move up where it should be)

Here's what I don't understand. What do I do if I have an N left over when I get this far?
 
Last edited:
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  • #2
After using ration test i get

lim x^(n+1)/(n+1)! *(n)/(x^n)! = (x)/(n+1)

ration of convergence is then n-> infinite |x|/n+1 -> 0

so for every chosen x you will get convergence Then R=infinite

Garret
-------------
Imagination is more important than knowlegde - A.Einstein
 
  • #3
Thanks man. I just got off the phone with my friend and he told me the same thing. I'm not sure waht I was thinking...it's been a long day.
 

1. What is a power series?

A power series is an infinite series of the form ∑ n=0 cn(x-a)n, where cn are constants and x is the variable. It is a type of mathematical series used to represent a function as a sum of infinitely many terms.

2. How do you find the radius of convergence of a power series?

The radius of convergence of a power series is the distance from the center of the series (the value of 'a') to the nearest point where the series converges. To find the radius of convergence, you can use the ratio test or the root test, which both involve taking the limit of the absolute value of the ratio or root of the coefficients in the power series.

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

The radius of convergence is important because it determines the values of x for which the power series will converge. If x is within the radius of convergence, the power series will converge and represent the function. If x is outside the radius of convergence, the series will diverge and not represent the function.

4. Can the radius of convergence of a power series be infinite?

Yes, it is possible for the radius of convergence to be infinite. This means that the power series will converge for all values of x. However, this is not always the case and the radius of convergence can also be a finite number or even 0, indicating that the series only converges at a single point.

5. How can power series be used to approximate functions?

Power series can be used to approximate functions by taking a finite number of terms in the series. The more terms that are included, the closer the approximation will be to the actual function. This is useful in mathematics and physics, where it is often easier to work with a power series than the original function.

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