Master Power Series Convergence with Expert Help - Examples Included

[PhysicsCollegeGirl]

Homework Statement

[/B]
There are three problems that I am struggling with.

1. ∑[k²(x-2)^k]/[3^k]
2. ∑[(x-4)ⁿ]/[(n)(-9)ⁿ]
3. ∑[2^k(x-3)^k]/[k(k+1)]

The Attempt at a Solution

On the first two I am having problems finding the end-points of the interval of convergence. I use the ratio test.

1. ∑[k²(x-2)^k]/[3^k]
Here I get I = (1,3)

2. ∑[(x-4)ⁿ]/[(n)(-9)ⁿ]
I = (-13,5)

And in the last one I got the end-points but I am not sure why the left one would diverge.
It says it diverges in the answers but I cannot see it.

3. ∑[2^k(x-3)^k]/[k(k+1)]

I = [5/2,7/2)

I use the alternating series test to evaluate x=5/2 and it doesn't seem to go to zero. I do l'hospital's, but it always shows me that it diverges.

 

[FactChecker]

Homework Statement

[/B]
There are three problems that I am struggling with.

1. ∑[k²(x-2)^k]/[3^k]
2. ∑[(x-4)ⁿ]/[(n)(-9)ⁿ]
3. ∑[2^k(x-3)^k]/[k(k+1)]

The Attempt at a Solution

On the first two I am having problems finding the end-points of the interval of convergence. I use the ratio test.

1. ∑[k²(x-2)^k]/[3^k]
Here I get I = (1,3)

2. ∑[(x-4)ⁿ]/[(n)(-9)ⁿ]
I = (-13,5)

You can tell immediately that this is wrong. The interval of convergence of this power series is centered at x=4. I could believe (3,5) but not (-13,5). (I don't know about the endpoints.)

And in the last one I got the end-points but I am not sure why the left one would diverge.
It says it diverges in the answers but I cannot see it.

3. ∑[2^k(x-3)^k]/[k(k+1)]

I = [5/2,7/2)

I use the alternating series test to evaluate x=5/2 and it doesn't seem to go to zero.

Look at this again. The terms are (-1)^k/[k(k+1)] at x=5/2.

 

[vela]

1. ∑[k²(x-2)^k]/[3^k]
Here I get I = (1,3)

It would be really helpful if you showed your work. Simply writing down the wrong answer doesn't give us much to go on to figure out where your mistakes are.