Solving Infinite Sums/Series: Kid Tutoring Homework

[muso07]

Homework Statement

Figuring out an infinite sum question for the kid I tutor... it's late and my brain's not functioning well. I think I'm overcomplicating the question, but I can't figure it out.

The the sum of the following infinite series:
S = 1+3x+5x^2+...

Homework Equations

The Attempt at a Solution

I got that tn = $$(2n-1)x^{n-1}$$

Which means the sum is
$$\Sigma (2n-1)x^{n-1}=2\Sigma nx^{n-1} - \Sigma x^{n-1} = 2\Sigma nx^{n-1} - \frac{1}{1-x}$$

And here is where I get stuck...

 

[tiny-tim]

Hi muso07!

Hint (for ∑ nx^n-1):

integrate. ​

 

[muso07]

Hm, the kid's in Year 11 so he hasn't done integration yet...

But when you integrate, you get x^n. Then $$lim_{(a\rightarrow\infty)} [x^{n}]^{a}_{1}= lim_{(a\rightarrow\infty)}x^{a}-x=-x$$
This isn't right, though... I feel like such an idiot. :P

 

[tiny-tim]

(have a sigma: ∑ and an infinity: ∞ and try using the X² and X₂ tags just above the Reply box )

No, you want d/dx lim_a->∞ (∑xⁿ)

 

[muso07]

You beat me to replying. Is the sum (1+x)/(1-x)^2?

 

[muso07]

Nevermind, I'm just going to tell him to believe me and it shall be great.

Thanks for all your help!