# Finding power series for given values of a sum

• Apr 29th 2011, 12:40 PM
Ulysses
Finding power series for given values of a sum
I have this exercise which I'm not sure how to solve.
It says: Consider the series $\displaystyle\sum_{n=0}^{\infty}x^n$ 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.
• Apr 29th 2011, 12:48 PM
girdav
Did you try to compute the sum?
• Apr 29th 2011, 12:51 PM
Ulysses
I'll try.
• Apr 29th 2011, 01:09 PM
Plato
Quote:

Originally Posted by Ulysses
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?

Hint. If the series converges $\sum\limits_{n = 0}^\infty {x^n } = \frac{1}{{1 - x}}$.
• Apr 29th 2011, 01:28 PM
topsquark
Quote:

Originally Posted by Ulysses
$\displaystyle\sum_{0}^{\infty}x^n$

Just as a side note for notation... I doubt anyone seriously thinks that this sum is over x. However you have not specified that. Here is the code to make this a sum over n from 0 to infinity:
\displaystyle\sum_{n = 0}^{\infty}x^n

-Dan
• Apr 29th 2011, 01:32 PM
chisigma
Finding the sum of the 'geometric series'...

http://quicklatex.com/cache3/ql_800e...2674717_l3.png

... is based on the 'algebraic identity'...

http://quicklatex.com/cache3/ql_3ee0...d348a49_l3.png (1)

... from which follows...

http://quicklatex.com/cache3/ql_e7b5...f745ab1_l3.png (2)

... so that if |x|<1 is...

http://quicklatex.com/cache3/ql_14ab...bbdd6e4_l3.png (3)

Kind regards

$\chi$ $\sigma$