Find the Interval of Convergence

[whatlifeforme]

Homework Statement

Find the interval of convergence.

Homework Equations

$\displaystyle \sum^{∞}_{n=0} \frac{(x-5)^n}{n^4 * 2^n}$

The Attempt at a Solution

I used the ratio test as follows:

$\displaystyle \frac{(x-5)^{n+1}}{(n+1)^4 * 2^{n+1}} * \frac{n^4 2^n}{(x-5)^n}$

taking the limit:

(x-5) lim (x->inf) $\displaystyle \frac{n^4}{2(n+1)^4}$ = 0

Most of these problems the limit comes to 1. How do i solve this now that the limit is zero?

 

[CompuChip]

You are very much on the right track, except that
$$\lim_{n \to \infty} \frac{n^4}{(n + 1)^4} \neq 0$$

You can see this if you expand $(n + 1)^4 = n^4 + \mathcal O(n^3)$

(Don't forget the factor of 1/2 in the end)

 

[whatlifeforme]

|x-5| < 1

-1 < x-5 < 1
4 < x < 6

Answer (interval of convergence): 4 < x < 6

Does that look right?

 

[whatlifeforme]

i think i forgot the factor of (1/2), right?

 

[HallsofIvy]

Yes, you did. The correct limit is 1/2. Now, what does that tell you about the radius of convergence?

 

[whatlifeforme]

Yes, you did. The correct limit is 1/2. Now, what does that tell you about the radius of convergence?

what is the radius of convergence vs interval of convergence?

 

[whatlifeforme]

|x-5| < 1

-1 < x-5 < 1
4 < x < 6

Answer (interval of convergence): 4 < x < 6

Does that look right?

is this part correct? because that is one of the choices as the answer but 2 < x < 3 is not.

 

[whatlifeforme]

bump.

 

[CompuChip]

what is the radius of convergence vs interval of convergence?

The radius of convergence is the "width" of the interval of convergence. E.g. if you are looking at the series around x = 2, if the radius of convergence is 4 the interval of convergence would be 2 - 4 < x < 2 + 4, i.e. -2 < x < 6.

Yes, you did. The correct limit is 1/2. Now, what does that tell you about the radius of convergence?

First try answering Halls' question. The radius of convergence is related to the ratio (1/2) you calculated.

bump.

Please don't do that, we are not online 24 hours per day and we will get back to you when we have time.

 

[whatlifeforme]

sorry for the bump. I'm getting 3 < x < 7 now that I added in the factor of 1/2

-1 < (x-5)/2 < 1