Finding the Sum of a Geometric Series with Infinite Terms

[dmitriylm]

Homework Statement

Find the sum of the series:

(1/(4^n))+ (((-1)^n)/(3^n))

from n=0 to infinity

Homework Equations

The Attempt at a Solution

I'm not overly familiar with series and am not sure how to approach this. A lot of help guides online talk about testing for divergence/convergence but how do I go about finding the sum?

 

[Hootenanny]

Homework Statement

Find the sum of the series:

(1/(4^n))+ (((-1)^n)/(3^n))

from n=0 to infinity

Homework Equations

The Attempt at a Solution

I'm not overly familiar with series and am not sure how to approach this. A lot of help guides online talk about testing for divergence/convergence but how do I go about finding the sum?

There is no general algorithm for finding the sum of a series. Indeed, it is often the case that series do not converge to something "nice". Whether you can write down a "nice" form for the sum usually depends on whether you can "spot" what is happening. In this case, it is fairly easy to spot the patterns. The first thing to note is that you can decompose your series into two sum:

\sum_{n=0}^\infty\left( \frac{1}{4^n} + \frac{(-1)^n}{3^n} \right) = \sum_{n=0}^\infty \frac{1}{4^n} + \sum_{n=0}^\infty \frac{(-1)^n}{3^n}.

Now you can deal with each series individually. Next, write down the first few terms of the two series and see if you can spot the pattern.

 

[dmitriylm]

I see the first sequence has a pattern of 1/(4^n) and the second one has a pattern of 1/(-3(^n)). Once I see the pattern what do I do?

 

[Hootenanny]

I see the first sequence has a pattern of 1/(4^n) and the second one has a pattern of 1/(-3(^n)). Once I see the pattern what do I do?

Can you see that each series tends to a particular number?

 

[dmitriylm]

Can you see that each series tends to a particular number?

I'm not clear on this, they are getting smaller so they would be getting closer to zero?

 

[Hootenanny]

I'm not clear on this, they are getting smaller so they would be getting closer to zero?

Yes, each term is getting progressively closer to zero - so what is happening to the sum? Perhaps it would help if you plotted the sum.

 

[dmitriylm]

I know using a calculator that the sum is a little over 2, but I would like to know how to do this on my own.

 

[HallsofIvy]

Those are both geometric series and there is a standard formula for the sum of a geometric series:
\sum_{n=0}^\infty r^n= \frac{1}{1- r}