How to Solve for n in a Power Series Limit?

In summary, the conversation discusses finding the limit of a power series, specifically when trying to cancel the n^n term on the denominator. The conversation includes a step-by-step solution and clarification on how to proceed with the problem.
  • #1
JRangel42
17
0
Power Series:(n^n)*(x^n)

Homework Statement



The only step I'm having problem with on this problem is when I take the Lim n→∞ of the problem. I want to know how to cancel the n^n on the denominator during one of the steps.

Homework Equations



∞
Ʃ (n^n)*(x^n)
n=1

The Attempt at a Solution



lim n→∞ [(n+1)^(n+1)]*[x^(n+1)][itex]/[/itex][(n^n)*(x^n)]

|x| lim n→∞ [(n+1)^(n+1)][itex]/[/itex]n^n

That's where I got stuck.
 
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  • #2


(n+1)n+1 / nn = ( (n+1)n / nn ) * (n+1). Can you see the limit now?
 
  • #3


Oh, I definitely see it now! Thanks. (^o^)/
 
  • #4


Can you show (n+1)^(n+1)/n^n>(n+1)??
 
  • #5


Yeah, I can definitely do that part, I just I had a little trouble looking for the one little section I had trouble with.
 

FAQ: How to Solve for n in a Power Series Limit?

What is a power series?

A power series is an infinite series of the form ∑n=0∞ cn(x-a)n, where cn and a are constants. It is a mathematical representation of a function as a sum of powers of the independent variable x.

What is the general term of a power series?

The general term of a power series is cn(x-a)n, where cn represents the coefficient of the nth power and a is the center of the series.

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

The radius of convergence, denoted by R, can be found using the ratio test: R = limn→∞ |cn+1/cn|. If the limit is zero, the series converges for all values of x. If the limit is infinity, the series converges only at x = a. If the limit is a finite number, the series converges for all values of x within the interval |x-a| < R.

What is the significance of the radius of convergence?

The radius of convergence determines the interval of values for which the power series converges. If x is outside this interval, the series will diverge. It also determines the interval of values for which the power series can be used to approximate the original function.

How do you calculate the sum of a power series?

The sum of a power series is the function that the series represents. To calculate the sum, you can either use the general term of the series or the coefficients and the center to write out the series and simplify it to a closed form expression.

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