An advanced factoring technique allows you to rewrite the sum of a finite geometric series in a compact formula. An infinite geometric series is more difficult because sometimes it sums to be a number and sometimes the sum keeps on growing to infinity. When does an infinite geometric series sum to be just a number and when does it sum to be infinity?
A geometric series is a sum of numbers whose consecutive terms form a geometric sequence. Recall the advanced factoring technique for the difference of two squares and, more generally, two terms of any power (5 in this case).
This is the sum of a finite geometric series.
As you can see, even numbers close to one either get very small quickly or very large quickly.
The formula for calculating the sum of an infinite geometric series that converges is:
A partial sum of an infinite sum is the sum of all the terms up to a certain point. Considering partial sums can be useful when analyzing infinite sums.
Compute the sum of the following infinite geometric series two ways, without using the infinite summation formula and using the infinite summation formula.
Without using the summation formula:
With using the summation formula:
What is the sum of the first 8 terms in the following geometric series?
You put $100 in a bank account at the end of every year for 10 years. The account earns 6% interest. How much do you have total at the end of 10 years?
Note that normally geometric series are written in the opposite order so you can identify the starting term and the common ratio more easily.
The sum of the 10 years of deposits is:
Find the sum of the first 15 terms for each geometric sequence below.
For each infinite geometric series, identify whether the series is convergent or divergent. If convergent, find the number where the sum converges.
12. You put $5000 in a bank account at the end of every year for 30 years. The account earns 2% interest. How much do you have total at the end of 30 years?
13. You put $300 in a bank account at the end of every year for 15 years. The account earns 4% interest. How much do you have total at the end of 10 years?
14. You put $10,000 in a bank account at the end of every year for 12 years. The account earns 3.5% interest. How much do you have total at the end of 12 years?
15. Why don’t infinite arithmetic series converge?
To see the Review answers, open this PDF file and look for section 12.5.