An infinite geometric series has an infinite amount of terms:
The sum of the series converges towards a particular number if the quotient is between and . In that case, the sum is
Example 1
An endowment is generated by a retired multimillionaire which is giving out an annual scholarship of to good students of mathematics, for all eternity. The money is deposited into a savings account with an annual interest rate of . What amount of money needs to be deposited into this account?
The present values of the annual payments to the mathematics student forms the infinite geometric series
where and . You know that the series is converging because is between and .
You have to find the sum of the an infinite geometric series to figure out how much that was deposited into the bank account to begin with:
When the quotient is a function of , the convergence area is given by . Then you can find the convergence area either by solving
or by solving
The two procedures are equal.
Rule
When the quotient is a function of , the convergence area is given by . Then you can find the convergence area by solving
Example 2
Option 1: Solving with two cases of inequality
You have a geometric series with the quotient . Find the area where it converges.
You begin by setting up the inequality:
Since this is an absolute value, you have to divide it into two inequalities. Solve them separately and use a sign chart to find the interval you are looking for.
From the sign lines you see can that the series converges when .
Example 3
Option 2: Solving with single inequality
You have a geometric series with the quotient .. Find the area where it converges.
You can also solve the exercise by looking at the inequality
In this case you get that
You can use the third algebraic identity to factorize the left-hand side:
Now, you can use a sign chart to find the answer. Draw them and interpret the lines:
Because you are looking for the area where , the answer is the interval with the dashed line. That gives you that the series converges when .