# Thread: Determining Value of a Series

1. ## Determining Value of a Series

We have started covering series and sequences in my calculus 2 class and I am having a hard time getting the problems started. For example I am told to determine the exact valiue of the following series

$\displaystyle \sum_{i\ =\ 0}^\infty 5 (\frac{2}{3})^i$

How do I proceed to start and complete the problem?

2. Originally Posted by fishguts
We have started covering series and sequences in my calculus 2 class and I am having a hard time getting the problems started. For example I am told to determine the exact valiue of the following series

$\displaystyle \sum_{i\ =\ 0}^\infty 5 (\frac{2}{3})^i$

How do I proceed to start and complete the problem?
You should have covered geometric series in your class. This is an infinite geometric series.

In general, a series of the form $\displaystyle \sum_{n=0}^\infty ar^n=\frac{a}{1-r}$ provided that $\displaystyle -1<r<1$

This should be all you need to complete the problem

3. ## Geometric Series

Hello fishguts
Originally Posted by fishguts
We have started covering series and sequences in my calculus 2 class and I am having a hard time getting the problems started. For example I am told to determine the exact valiue of the following series

$\displaystyle \sum_{i\ =\ 0}^\infty 5 (\frac{2}{3})^i$

How do I proceed to start and complete the problem?
I assume that you have covered Geometric Series, because this is one:

$\displaystyle \sum_{i\ =\ 0}^\infty 5 (\frac{2}{3})^i=5 + 5.\tfrac23 + 5.(\tfrac23)^2+5.(\tfrac23)^3+...$

which has first term $\displaystyle a = 5$, common ratio $\displaystyle r = \tfrac23$

The sum to infinity is $\displaystyle \frac{a}{1-r}$.

Can you complete it now?

4. For geometric series with common ratio r<1,

$\displaystyle \sum^{\infty}_{0} ar^{i} = \frac {a}{1-r}$

5. so then after plugging in all the numbers I get 15. Is this correct? it seems too easy...

6. Hello fishguts
Originally Posted by fishguts
so then after plugging in all the numbers I get 15. Is this correct? it seems too easy...
Yep! That's all there is to it. 15 is the answer.