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Expert: Steve Holleran - 5/13/2008


Question

Geometric Sequence
I'm studying for my final exam on Friday, and my review packet has a problem that I can't remember how to solve and for which I can't seem to find any old worksheets that show me how to do it.

It says: "Evaluate the infinite geometric series:"
(Forgive me if this gets convoluted but it's a pretty funny-looking problem)

Additionally, there's a problem that says:
"The initial swing of a pendulum is 14 inches. Each successive swing is 0.8 times the previous swing. When it stops, how far will the pendulum have swung?"
I realize there's a logic-based approach that can be taken to this, but I think I'm supposed to find the answer using a series of some kind. I'm so foggy on the subject...Any help would be GREATLY appreciated!

And lastly (sorry!)
When does an infinite sum exist?
What is the geometric sum formula?
What is a common ratio?

I've been trying to look this up on other websites but they all seem to be providing answers much more complicated than what I need or can understand. Again, thanks to whomever answers!

Answer
Hi Maria,

Let's see if I can put all this in a context that will help you prepare for your exam.  I'll answer a little out of the order that you presented the problems.

First of all, let's address your last concerns:

In a geometric series, finite or infinite, the common ratio is the multiplier used to get each succeeding term.  Or, you can think of it as any term divided by the previous one.  For example, in the series

     1 + 1/2 + 1/4 + ...    the common ratio is 1/2

In the series 1 + 3 + 9 + 27 + ...    the ratio is 3.  
Remember that to find it, all you have to do is divide ANY term by the term right before it.

Now, an INFINITE geometric series will have a sum IF its common ratio is  less than 1 (positive or negative). So a series with a ratio of 2/3 will have a sum (converge) and a series with a ratio of - 5/4 will not have a sum (diverge).

If an infinite geometric series has a ratio whose absolute value is less than 1, and so has a sum, the formula is

             S(inf) = a / (1-r)   where S(inf) just stands for the

sum of an infinite series, a is the first term, and r is the common ratio.

Getting back to your first question, then, if you write out a few terms of this series, you'll get

        8 + 8/3 + 8/9 + 8/27 + ...  

I think you can see that the ratio is r = 1/3.  (Notice its what's being raised to the power in the sigma notation.).

Okay, so then this series has a sum , and its :

                S(inf) = 8 / (1 - 1/3) = 8 / (2/3) = 12.


For the pendulum problem, you have a similar situation.  The first term is 14, then to get each succeeding term, you have to multiply by 0.8:

 14 + 14(0.8) + 14(0.8^2) + 14(0.8^3) + ...

So here the ratio is 0.8.  You assume infinite swings to get it to come to rest.  The sum would be

              S(inf) = 14 / (1-0.8) = 14 / 0.2 = 70.

I hope this helps you get ready for your test, and I hope its clear.
Good luck!

Steve
    Questioner's Rating
    Rating(1-10)Knowledgeability = 10Clarity of Response = 10Politeness = 10
    CommentFANTASTIC. This is so much better. 10/10 for everything, can't thank you enough!


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Steve Holleran

Expertise

I can help with all math questions from basic math to Calculus. Whether it`s consumer questions, or questions from high school or college students, I have probably dealt with it at some time in my career.

Experience

33 years teaching experience in NJ public schools

Education/Credentials
B.S. Mathematics : Wake Forest University 1972 M.S. Mathematics : Monmouth University 1981

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