What is the Interval of Convergence for These Power Series?

In summary, the first problem involves finding the interval of convergence for a series involving x and n. The answer is -8<x<-6. The second problem involves finding the interval of convergence for a series involving (-1)^n, x, and n!. The answer is (-\infty, \infty). There was some confusion about the solution, but it was clarified.
  • #1
fsm
88
0
I just wanted to see if someone could verify my answers:

[tex]
\\sum_{n=0}^\\infty \\frac{(x+7)^n}{sqrt(n)}
[/tex]
I get:
-8<x<-6

[tex]
\\sum_{n=1}^\\infty \\frac{(-1)^n*x^2n}{n!}
[/tex]
This one I'm not sure of. When I take the limit I get 0. When I solve the inequality I get x. I can't find an example of this.
 
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  • #2
This might be better:
1.gif
 
  • #3
1. correct
2. [tex]\sum_{n=1}^{\infty} \frac{(-1)^{n}x^{2n}}{n!} [/tex][tex] |\frac{(-1)^{n+1}x^{2n+1}}{n!(n+1)}\frac{n!}{(-1)^{n}x^{2n}} = \frac{x}{n+1} \rightarrow 0 [/tex]. Thus the interval of convergence is [tex] (-\infty, \infty) [/tex]
 
Last edited:
  • #4
I don't understand your answer for #2. I got the same thing but how is this the interval of convergence?
 
  • #5
fsm said:
I don't understand your answer for #2. I got the same thing but how is this the interval of convergence?

What don't you understand about it?
 
  • #6
All the stuff to the right of the equal sign now just appeared. Thanks for the help.
 

What is an interval of convergence?

An interval of convergence is a range of values for which a power series will converge, meaning the sum of the terms in the series will approach a finite value as the number of terms increases.

How is the interval of convergence determined?

The interval of convergence is determined by using the ratio test, which compares the absolute value of a term in the series to the absolute value of the next term. If the ratio of these values is less than 1, the series will converge within that range.

What happens if the ratio test is inconclusive?

If the ratio test is inconclusive, other tests such as the root test or the integral test can be used to determine the interval of convergence.

Can the interval of convergence be infinite?

Yes, the interval of convergence can be infinite, meaning the series will converge for all values of the variable. This is known as a radius of convergence of infinity.

What are the implications of the interval of convergence?

The interval of convergence is important for determining the validity and behavior of a power series. It also allows for the approximation of functions using the power series within the interval of convergence.

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