SOLUTION: Please help me solve this problem. It says, Decide whether each infinite geometric series diverges or converges. State whether each series has a sum. I understand that it diver

Question 196526: Please help me solve this problem.
It says, Decide whether each infinite geometric series diverges or converges. State whether each series has a sum.
I understand that it diverges when the absolute value of r is > or = 1 and converges when the absolute value of r is < 1, but how do you know whether each series has a sum. For example one problem says:
1. 4 + 2 + 1 + ...
and I know that r=2 therefore it diverges but I can not figure out whether it has a sum or not.
Thanks for your time.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Well first off, if a series diverges, it does not have a sum (ie the if you add up the infinite terms, you won't get a fixed number).

In this problem, however, the series does converge. Here's why:

The sequence of terms: 4, 2, 1, ... can be modeled by the formula:

where "n" starts at (note: plug in some values to test this)

Since this equation fits the form and , this means that the absolute value of "r" is less than 1. So the series converges.

To find the sum of the infinite series, simply use the formula where "a" is the first term and "r" is given as

So the infinite sum is...

In other words,

4+2+1+...=8

I hope this makes sense.
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