Sum to infinity

In a nutshell

The sum to infinity of a geometric series can be found using a formula if the series is convergent.

Calculating the sum to infinity

In a convergent geometric series, as $$n$$ approaches infinity, the sum $$S_n$$ will approach a finite value. This means that a sum can be calculated. A geometric series is convergent when $$\lvert r \rvert < 1$$ where $$r$$​ is its common ratio.

​$$\boxed {S_\infin = \dfrac {a}{1-r}}$$

Deriving the formula

The formula for a geometric series for a finite value of $$n$$: ​

$$S_n = \dfrac{a(1-r^n)}{1-r}$$​​

A geometric series is convergent when $$\lvert r \rvert < 1$$​. Considering this as $$n$$ becomes larger towards: 

​$$r^n \rightarrow 0$$​​

This means when calculating the sum to infinity:

$$S_\infin = \dfrac{a(1-r^n)}{1-r} \rightarrow \dfrac{a(1-0)}{1-r} \rightarrow \dfrac{a}{1-r}$$​​

Example 1

A geometric sequence has a third term of $$0.25$$ and sixth term of $$0.03125$$. First show that this series is convergent, then calculate $$S_\infin$$.​

Use the $$n^{th}$$ term to form simultaneous equations:

$$\begin{aligned} ar^{n-1} &= ar^2 = 0.25 \\ ar^{n-1} &= ar^5 = 0.03125\end{aligned}$$​​

Solve for $$r$$:

​$$\dfrac {ar^5}{ar^2} = r^3 = \dfrac {0.03125}{0.25}$$​​

​$$r^3 = 0.125$$​​

​$$\underline {r = \sqrt[3]{0.125} = 0.5}$$​​

The series is convergent because $$\underline{r<1}$$.

Solve for $$a$$:

$$a(0.5)^2 = 0.25$$​​

​$$a = \dfrac {0.25}{0.5^2} = 1$$​​

Find $$S_\infin$$:

​$$S_\infin = \dfrac {a}{1-r} = \dfrac {1}{1-0.5} = \dfrac{1}{0.5}$$​​

​$$\underline{S_\infin = 2}$$​​

Note: Each term in this sequence is the previous term halved, starting from $$1$$. Adding ad infinitum gives $$2$$.