Geometric series

 Definition. A geometric series is a series in the form:

a is called the initial term because it is the first term, r is called the ratio.

Example: 

is a series with initial term 1 and ration r = 1/2

Convergence and Divergence

In the geometric series having the form above

If ❘r❘ <1, the series converges. The series can be written as:

It  converges to a/1-r

If ❘r❘≥1, the series diverges.

Exercises

Determine whether the following series converges or diverges. If it converges find its sum:

Solutions

a) Let's find the first several terms in the series:

 

 

The initial term is the first term. We can see that it's repeating in the following term as a product of itself by the power of another term. The initial term here is a = 9 and the ratio is -3/4. Since ❘-3/4❘<1, the series converges. It converges to a/1-r = 9/1-(-3/4) = 9/1+3/4 = 9 /7 over 4 = 36/7. The series converges to 36/7

b) 

Practice