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Sum to infinity

Sum to infinity

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Explainer Video

Tutor: Labib

Summary

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 nnn approaches infinity, the sum SnS_nSn​ will approach a finite value. This means that a sum can be calculated. A geometric series is convergent when ∣r∣<1\lvert r \rvert < 1∣r∣<1 where rrr​ is its common ratio.


​S∞=a1−r\boxed {S_\infin = \dfrac {a}{1-r}}S∞​=1−ra​​


Deriving the formula

The formula for a geometric series for a finite value of nnn: ​

Sn=a(1−rn)1−rS_n = \dfrac{a(1-r^n)}{1-r}Sn​=1−ra(1−rn)​​​


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

​rn→0r^n \rightarrow 0rn→0​​


This means when calculating the sum to infinity:

S∞=a(1−rn)1−r→a(1−0)1−r→a1−rS_\infin = \dfrac{a(1-r^n)}{1-r} \rightarrow \dfrac{a(1-0)}{1-r} \rightarrow \dfrac{a}{1-r}S∞​=1−ra(1−rn)​→1−ra(1−0)​→1−ra​​​


Example 1

A geometric sequence has a third term of 0.250.250.25 and sixth term of 0.031250.031250.03125. First show that this series is convergent, then calculate S∞S_\infinS∞​.​


Use the nthn^{th}nth term to form simultaneous equations:

arn−1=ar2=0.25arn−1=ar5=0.03125\begin{aligned} ar^{n-1} &= ar^2 = 0.25 \\ ar^{n-1} &= ar^5 = 0.03125\end{aligned} arn−1arn−1​=ar2=0.25=ar5=0.03125​​​


Solve for rrr:

​ar5ar2=r3=0.031250.25\dfrac {ar^5}{ar^2} = r^3 = \dfrac {0.03125}{0.25}ar2ar5​=r3=0.250.03125​​​

​r3=0.125r^3 = 0.125r3=0.125​​

​r=0.1253=0.5‾\underline {r = \sqrt[3]{0.125} = 0.5}r=30.125​=0.5​​​


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


Solve for aaa:

a(0.5)2=0.25a(0.5)^2 = 0.25a(0.5)2=0.25​​

​a=0.250.52=1a = \dfrac {0.25}{0.5^2} = 1a=0.520.25​=1​​


Find S∞S_\infinS∞​:

​S∞=a1−r=11−0.5=10.5S_\infin = \dfrac {a}{1-r} = \dfrac {1}{1-0.5} = \dfrac{1}{0.5}S∞​=1−ra​=1−0.51​=0.51​​​

​S∞=2‾\underline{S_\infin = 2}S∞​=2​​​


Note: Each term in this sequence is the previous term halved, starting from 111. Adding ad infinitum gives 222.

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FAQs - Frequently Asked Questions

How do you calculate the sum to infinity of a convergent geometric series?

By using the formula a/(1-r) when r is between -1 and 1.

When is a geometric series convergent?

When the common ratio, r, is between -1 and 1.

What type of series will have a finite sum when all its terms are added together?

A convergent series.

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