Math · Calculus II · Worked example

Sum a geometric series

Find the sum of 3 + 3/2 + 3/4 + 3/8 + ⋯.

Find the ratio

Each term is half the one before: the first term is a = 3 and the ratio is r = 1/2.

Check that it converges

|r| = 1/2 < 1, so the series converges.

Sum it

Divide the first term by 1 − r.

\frac{3}{1 - \frac{1}{2}} = 6

Compare with the partial sums

Ten terms give 3069/512 = 5.994. Each gap to 6 is half the one before, so the partial sums close in on 6.

\frac{3 (1 - {(\frac{1}{2})}^{10})}{1 - \frac{1}{2}} = \frac{3069}{512}

Result

The sum is 6.

Your turn

Find the sum of 1 − 2/3 + 4/9 − 8/27 + ⋯.

Show the answer and explanation

3/5.

It is geometric with a = 1 and r = −2/3, and |r| < 1, so the sum is 1/(1 − (−2/3)) = 3/5.

\frac{1}{1 - (- \frac{2}{3})} = \frac{3}{5}

Keep exploring

In Sequences & infinite series, change the ratio to 2. The terms grow and the series diverges: the formula 3/(1 − r) would give −3, which is not a sum at all.

Return to the concept →

Sources and scope

Authored study material. Tool results depend on the stated inputs and model assumptions.

Make it concrete

Try in the workspace

Open the example inputs, change a value and keep a useful result on your board.

Sum it in Sequences & infinite series Check the sum in Math Open worked example on a board Infinite geometric sum in Math Reference

Your existing work stays on this device. Examples open as editable copies.