Geometric Series (College Board AP® Calculus BC): Study Guide

Geometric series

What is a geometric series?

• A geometric series is a series of the form

  • where is a real number constant

    • is also the first term of the series

  • and is a real number known as the common ratio

  • Note that the series starts with

• Note that there is a constant ratio between one term and the next

  • For example

    • , , , etc.

    • Each term is the preceding term multiplied by the common ratio

When does a geometric series converge?

• The sum of the first terms of a geometric series is given by

  • 

    • I.e.

    • 

    • etc.

  • If , then the series is so

• If then the geometric series converges

  • and

    • This comes from

• If then the geometric series diverges

  • For example if , then the series is

    • The sequence of partial sums is , which diverges to (depending on whether is positive or negative)

    • So the series diverges

  • Or if and , then the series is

    • Each term added on is bigger than the one before

    • The sequence of partial sums is , which diverges to

    • So the series diverges

Alternative forms of a geometric series

How else can a geometric series be written?

• You may sometimes see geometric series written with the sum starting at

• One example is

  • In this form most things remain the same

    • is still the first term of the series

    • is still the common ratio

    • The infinite sum still converges to if

  • But the formula for is slightly different

    • gives the sum of the first terms of the series

• Another example is

  • In this form some things remain the same

    • is still the common ratio

    • The infinite sum still only converges if

  • But some other things change

    • is the first term of the series

    • The infinite sum, if it converges, converges to

    • gives the sum of the first terms of the series

• You can work with these forms if you prefer

  • unless an exam question specifically indicates otherwise