Infinite series problems

[Cap'n Refsmmat]

(This would ordinarily be in Homework Help, but we are not actually graded on homework and it fits better here.)

 

I have a problem which looks like this:

 

Find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x.

 

[math]\sum^{\infty}_{n = 1} \frac{x^n}{2^n}[/math]

 

I don't really know where to start. Intuition tells me I could just declare x < 2, but I do not know how to go about finding the actual sum of the series or proving that my intuition is actually correct.

 

I'd imagine I am supposed to create a sequence of partial sums and see if the sequence converges (via a limit), but how would I do that for this function?

[the tree]

The Wikipedia page on geometric series gives you the actual sum. I'd imagine you could take it as a given that [imath]\sum r^n[/imath] converges for [imath]|r|<1[/imath] but that's a question of how much detail is required. Take care with exactly how you write the inequalities - negatives [imath]x[/imath]'s should be included as well.

[Cap'n Refsmmat]

Oh, right, I forgot about rewriting it as

 

[math]\sum_{n = 1}^{\infty} \left(\frac{x}{2} \right )^n[/math]

 

Oops. Thanks for the moment of inspiration.