Find the power series for representatino for the function

[vande060]

Homework Statement

Ive been able to do every single problem in my homework to the point where I have to test the edges of the interval of convergence. I have not been able to figure out a single one of the problems at the point of testing the edge of convergence, and I am to the point of pulling my hair out, so I really need an explanation of what to do at that point. Ill give an example of a problem.find the power series for representation for the function

f(x) = x/(9+x^2)

Homework Equations

The Attempt at a Solution

f(x) = x/(9+x²)

f(x) = x/(9 -(-x²)

f(x) = x/9(1-(-x²/9)

(∞, n=0) (x/9)∑ (-1)ⁿ [(x²)/(9)] ⁿ

(∞, n=0) ∑ (-1)ⁿ (x²ⁿ⁺¹)/ (9ⁿ⁺¹)

then i do a quicktest to find the radius of convergence

lim _{n--> ∞} |c_n/c_n+1|

lim _{n--> ∞} |[(-1)ⁿ/(9ⁿ⁺¹)] * [(9ⁿ⁺²)/(-1)ⁿ⁺¹] = 9

|x²| < |9|

-3 <x < 3

this is where i start to feel bad, i know the radius of convergence is 3 and the interval of convergence is (-3,3), but I have to test the edges of convergence, and I can't seem to do it. I know i should plug -3 and 3 back into the power series and test for convergence, but i have no idea how to test these two series for convergence

(∞, n=0) ∑ (-1)ⁿ ((-3)²ⁿ⁺¹)/ (9ⁿ⁺¹)

(∞, n=0) ∑ (-1)ⁿ ((3)²ⁿ⁺¹)/ (9ⁿ⁺¹)

 

[Dick]

What are 3^(2n) and (-3)^(2n) expressed as powers of 9?

 

[vande060]

What are 3^(2n) and (-3)^(2n) expressed as powers of 9?

9^n and 9^n(∞, n=0) ∑ (-1)ⁿ ((9)^n+1/2)/ (9ⁿ⁺¹)

which diverges by alternating series test because bn+1< bn and limit runs to 0, right?

so interval of convergence is (-3,3)

 

[Dick]

9^n and 9^n(∞, n=0) ∑ (-1)ⁿ ((9)^n+1/2)/ (9ⁿ⁺¹)

which diverges by alternating series test because bn+1< bn and limit runs to 0, right?

so interval of convergence is (-3,3)

Why don't you explicitly write out the first three or four terms in each series and then think about that again? You don't need an alternating series test to conclude that they diverge.

 

[vande060]

Why don't you explicitly write out the first three or four terms in each series and then think about that again? You don't need an alternating series test to conclude that they diverge.

unfortunately I do :( Professor said that writing out the first few terms of a series is not enough to draw conclusions. We have to use a test, and cite what test we used, for every problem. I get where you are coming from though, and thanks for the help, I cracked out a few more problems on my homework after working through this one :D

 

[Chantry]

Don't forget that for some problems you can differentiate or integrate the power series of a simpler function.

 

[Dick]

unfortunately I do :( Professor said that writing out the first few terms of a series is not enough to draw conclusions. We have to use a test, and cite what test we used, for every problem. I get where you are coming from though, and thanks for the help, I cracked out a few more problems on my homework after working through this one :D

My suggestion to write out a few terms isn't meant to be a proof. It's meant to help you get an idea of what kind of series you are dealing with. Then you prove it.