Determine if the following series is convergent, and if so, find its sum:
The sum of the series is therefore 3/5.
Explanation of Each Step
We first rewrite the problem so that the summation starts at one and is in the familiar form of a geometric series, whose general form is
After bringing the negative one and the three fifths together, we see that our given infinite series is geometric with common ratio -3/5.
For a geometric series to be convergent, its common ratio must be between -1 and +1, which it is, and so our infinite series is convergent.
We must now compute its sum.
The given series
starts the summation at , so we shift the index of summation by one:
Our sum is now in the form of a geometric series with a = 1, r = -2/3. Since |r| < 1, the series converges, and its sum is
In Step (3) we applied the formula for the sum of a geometric series:
This formula was derived in a previous section of this lesson.
Possible Challenge Areas
Students who may have been confused by this step may wish to refer to the previous lesson on Sigma Notation, where this process was explained in more detail.