# Closed Formula for Geometric Series and Sums

• Feb 23rd 2010, 05:55 PM
Closed Formula for Geometric Series and Sums
I just have two questions regarding closed form formulas for geometric series and summing infinite series.

For the closed form formula, I have an expression for the $\displaystyle nth$ term:

$\displaystyle 7(7/8)^n$

But I am not sure how to convert this to closed form since I am not really sure what closed form looks like. Is it just to convert it into a finite sum equation like this:

$\displaystyle \frac{7(1-\frac {7} {8}^n)} {(1- \frac {7} {8})}$

For the infinite series, I have the formula:

$\displaystyle \frac {1} {4} \sqrt{h}$

Where $\displaystyle h = 7(7/8)^n$

So the very first term would be:

$\displaystyle \frac {1} {4} \sqrt{7\frac {7} {8}}$

And this would go in the numerator, but I am not sure what would go in the denominator. I really appreciate everyone's help.
• Feb 23rd 2010, 06:00 PM
icemanfan
Quote:

Originally Posted by Spudwad
I just have two questions regarding closed form formulas for geometric series and summing infinite series.

For the closed form formula, I have an expression for the $\displaystyle nth$ term:

$\displaystyle 7(7/8)^n$

But I am not sure how to convert this to closed form since I am not really sure what closed form looks like. Is it just to convert it into a finite sum equation like this:

$\displaystyle \frac{7(1-\frac {7} {8}^n)} {(1- \frac {7} {8})}$

This is almost perfect, but remember, the first term of that series is not 7. it is $\displaystyle 7 \cdot \frac{7}{8}$.

Quote:

For the infinite series, I have the formula:

$\displaystyle \frac {1} {4} \sqrt{h}$

Where $\displaystyle h = 7(7/8)^n$

So the very first term would be:

$\displaystyle \frac {1} {4} \sqrt{7\frac {7} {8}}$

And this would go in the numerator, but I am not sure what would go in the denominator. I really appreciate everyone's help.
What's the common ratio? You're taking the square root of $\displaystyle \left(\frac{7}{8}\right)^n$, which is $\displaystyle \left(\frac{7}{8}\right)^{0.5 \cdot n} = \left(\left(\frac{7}{8}\right)^{0.5}\right)^n$.