Finding power series for given values of a sum

[Telemachus]

Homework Statement

I have this exercise which I'm not sure how to solve.
It says: Consider the series $$\displaystyle\sum_{0}^{\infty}x^n$$ Does exists any value of x for which the series converges to five? ¿and to 1/3?

Well, I've reasoned that if there exists that value, then it must be inside of the radius of convergence for the series. So I've found the radius of convergence:

$$a_n=1$$

$$R=\displaystyle\lim_{n \to{}\infty}{\left |{\displaystyle\frac{a_n}{a_{n+1}}}\right |}=1$$

But now I don't know how to proceed.

 

[Sethric]

First of all, your formula for the radius of convergence looks a bit off. Where is the 1/2 coming from? And where is the x? That said, finding the radius of convergence will only let you know what values of x will allow the series to converge, not what it will converge to.

Do you know what the geometric series is? When does it converge? What does it converge to?

 

[Telemachus]

Sorry, I've corrected it, I did $$a_{n+1}=2$$ but its 1, I've just corrected it.

Thanks.

 

[HallsofIvy]

[tex]\sum_{n= 0}^\infty x^n[/itex] is a geometric series. There is a simple formula for its sum. Do you know what it is?

 

[Telemachus]

Yes, thanks. I've found it. I didn't realized it was a geometric series because I haven't been dealing with series for a while, but it was easy :)