How to Find the Sum to Infinity of a Geometric Series

Sum to Infinity of a Geometric Series: Video Lesson

What is the Sum to Infinity?

The sum to infinity is the result of adding all of the terms in an infinite geometric series together. It is only possible to calculate the sum to infinity for geometric series that converge. This means that the size of each new term must be smaller than its previous term.

A geometric series is obtained when each term is multiplied by the same number from one term to the next. The value that each term is multiplied by to get to the next term is called the ratio.

For example, in the sequence 4+2+1+0.5+…, the terms are halving each time. Therefore r equals one half.

S sub n means the sum of the first ‘n’ terms.

For example, S sub 1 equals 4 and S sub 2 equals 4 plus 2 equals 6.

As more terms are added, we see that S sub 3 equals 7, S sub 4 equals 7.5, S sub 5 equals 7.75 and S sub 6 equals 7.875.

Because the terms are getting smaller and smaller, as we add more terms, we are adding an increasingly negligible amount.

Progressing further, S sub 7 equals 7.9375, S sub 8 equals 7.96875. We can see that the sum is approaching 8.

Even adding the first 20 terms, S sub 20 equals 7.99999237061.

Eventually, if an infinite number of terms could be added, the sum would indeed approach 8.

We say that the sum to infinity is 8, or S sub infinity equals 8

The sum to infinity of the series is calculated by S sub infinity equals the fraction with numerator a. sub 1 and denominator 1 minus r, where a. sub 1 is the first term and r is the ratio between each term.

For this series, where a. sub 1 equals 4 and r equals 0.5, S sub infinity equals the fraction with numerator 4 and denominator 1 minus 0.5 which becomes S sub infinity equals 4 over 0.5 equals 8.

The sum of an infinite number of terms of this series is 8.

This means that the sequence sum will approach a value of 8 but never quite get there.

How to Find the Sum to Infinity of a Geometric Series

The sum to infinity of a geometric series is given by the formula S∞=a1/(1-r), where a1 is the first term in the series and r is found by dividing any term by the term immediately before it.

  • a1 is the first term in the series
  • ‘r’ is the common ratio between each term in the series
the formula for the sum to infinity

S sub infinity equals the fraction with numerator a. sub 1 and denominator 1 minus r

The sum to infinity of a geometric series

To find the sum to infinity of a geometric series:

  1. Calculate r by dividing any term by the previous term.
  2. Find the first term, a1.
  3. Calculate the sum to infinity with S∞ = a1 ÷ (1-r).

For example, find the sum to infinity of the series 1 plus one third plus one ninth plus period period period

Step 1. Calculate r by dividing any term by the previous term

We can divide the term one third by the term before it, which is 1.

one third divided by 1 equals one third and so, r equals one third.

It does not matter which term you choose, simply divide any term by the term before it to find the value of r.

For example, the same result is obtained by considering the last two terms instead: one ninth divided by one third equals one third.

Step 2. Find the first term, a1

The first term is simply the first number in the series, which is 1.

Step 3. Calculate the sum to infinity with S∞ = a1 ÷ (1-r)

The sum to infinity is given by S sub infinity equals the fraction with numerator a. sub 1 and denominator 1 minus r.

a. sub 1 equals 1 and r equals one third.

Therefore the sum to infinity becomes S sub infinity equals the fraction with numerator 1 and denominator 1 minus one third which becomes S sub infinity equals 1 over two thirds. This simplifies to S sub infinity equals three halves or S sub infinity equals 1.5.

example of calculating the sum to infinity

When Does the Sum to Infinity Exist?

The sum to infinity only exists if -1<r<1. If the common ratio is outside of this range, then the series will diverge and the sum to infinity will not exist. If |r|<1, the sequence will converge to the sum to infinity given by S∞=a/(1-r).

A convergent geometric series is one in which the terms get smaller and smaller. This means that the terms being added to the total sum get increasingly small. The series converges to a final value.

For example, in the series one half plus one fourth plus one eighth plus period period period, the fractions can be seen to fit inside the area of a 1 by 1 square. Therefore the fractions one half plus one fourth plus one eighth plus period period period will fill an area of 1 u n i t s squared.

The series one half plus one fourth plus one eighth plus period period period converges to 1.

1/2+1/4+1/8 = 1 proof convergent series

The series converges because the terms are getting smaller in magnitude. We are adding less and less each time.

Geometric series converge and have a sum to infinity if |r|<1. The common ratio must be between -1 and 1.

A geometric series diverges and does not have a sum to infinity if |r|≥1. If the terms get larger as the series progresses, the series diverges.

The sum to infinity does not exist if |r|≥1.

For example, the series 1 plus 2 plus 4 plus 8 plus 16 plus 32 plus period period period is a divergent series because the terms get larger. The common ratio is 2 and a geometric series will diverge if |r|≥1.

example of a divergent series 1+2+4+8+...

For a series to converge, the terms must get smaller and smaller in magnitude as the series progresses.

For a geometric series, the series converges if |r|<1.

Arithmetic series do not converge and so they do not have a defined sum to infinity. If the common difference is positive, then the sum to infinity of an arithmetic series is +∞. If the common difference is negative, the sum to infinity is -∞.

Sum to Infinity Calculator

Enter the first two terms of a geometric sequence into the calculator below to calculate its sum to infinity.

Negative Sum to Infinity

The sum to infinity of a geometric series will be negative if the first term of the series is negative.

This is because the sum to infinity is given by S sub infinity equals the fraction with numerator a. sub 1 and denominator 1 minus r.

For a sum to infinity to exist, negative 1 is less than r is less than 1. This means that the denominator of the sum to infinity equation can never be negative.

The only way to obtain a negative sum to infinity is for the numerator, a1, to be negative.

a1 is the first term of the series. Hence, if the first term is negative, the sum to infinity will also be negative.

For example, find the sum to infinity of negative 8 minus 4 minus 2 minus period period period

Here, a. sub 1 equals negative 8 and r equals one half.

Therefore the sum to infinity S sub infinity equals the fraction with numerator a. sub 1 and denominator 1 minus r becomes S sub infinity equals the fraction with numerator negative 8 and denominator 1 minus one half which equates to S sub infinity equals negative 16.

Sum to Infinity of an Alternating Series

A geometric series will alternate between positive and negative terms if the ratio is negative.

For example, in the series negative 6 plus 2 minus two thirds plus period period period the terms alternate from negative to positive.

The ratio, r equals negative one third.

Since a. sub 1 equals negative 6, the sum to infinity S sub infinity equals the fraction with numerator a. sub 1 and denominator 1 minus r becomes S sub infinity equals the fraction with numerator negative 6 and denominator 1 minus negative one third.

The denominator simplifies to S sub infinity equals the fraction with numerator negative 6 and denominator 1 plus one third and this can be evaluated so that S sub infinity equals negative 4.5.

Examples of Calculating the Sum to Infinity

Here are sum examples of calculating the sum to infinity for geometric series.

In each case, the sum to infinity formula S sub infinity equals the fraction with numerator a. sub 1 and denominator 1 minus r will be used, where a1 is the first term and r is the ratio.

Geometric SeriesFirst Term, a1Ratio, rCalculationSum to Infinity
9 plus 3 plus 1 plus period period period9one thirdS sub infinity equals the fraction with numerator 9 and denominator 1 minus one thirdS sub infinity equals 13.5
5 plus 1 plus 0.2 plus period period period5one fifthS sub infinity equals the fraction with numerator 5 and denominator 1 minus one fifthS sub infinity equals 6.25
9 plus 6 plus 4 plus period period period9two thirdsS sub infinity equals the fraction with numerator 9 and denominator 1 minus two thirdsS sub infinity equals 27
20+2+0.2+…20one tenthS sub infinity equals the fraction with numerator 20 and denominator 1 minus one tenthS sub infinity equals the repeating decimal 2 2 point followed by repeating digit 2
4.8+1.2+0.34.8one fourthS sub infinity equals the fraction with numerator 4.8 and denominator 1 minus one fourthS sub infinity equals 6.4
4 plus 2 the square root of 2 plus 2 plus period period period4the fraction with numerator 1 and denominator the square root of 2S sub infinity equals the fraction with numerator 4 and denominator 1 minus the fraction with numerator 1 and denominator the square root of 2S sub infinity almost equals 13.66
negative 8 plus 6 minus 4.5 plus period period period-8negative three fourthsS sub infinity equals the fraction with numerator negative 8 and denominator 1 minus negative three fourthsS sub infinity almost equals negative 4.57

How to Write a Recurring Decimal as a Fraction with an Infinite Series

Recurring decimals can be written as a fraction using the geometric infinite series formula S∞=a/[1-r]. A decimal can be written as fractions out of 10, 100, 1000 and so on. Written in this way, the recurring decimal can be written as a geometric series in which the first term and ratio can be found.

Recurring Decimal to Fraction: Example 1

For example, write the repeating decimal 0 point followed by repeating digit 3 as a fraction.

the repeating decimal 0 point followed by repeating digit 3 equals 0.333333333 period period period

The recurring decimal can be written as a series of fractions out of 10, 100, 1000 and so on.

the repeating decimal 0 point followed by repeating digit 3 equals three tenths plus 3 over 100 plus 3 over 1000 plus period period period

The first term, a. sub 1 equals three tenths.

The ratio is r equals one tenth since three tenths is divided by 10 to make 3 over 100 and so on.

Therefore the sum to infinity S sub infinity equals the fraction with numerator a. sub 1 and denominator 1 minus r becomes S sub infinity equals the fraction with numerator three tenths and denominator 1 minus one tenth.

This simplifies so that S sub infinity equals the fraction with numerator three tenths and denominator nine tenths. Evaluating S sub infinity equals three tenths divided by nine tenths, the sum to infinity is S sub infinity equals three ninths.

Recurring Decimal to Fraction: Example 2

Write the recurring decimal the repeating decimal 0 point followed by repeating digits 1 2 as a fraction.

Here the repeating decimal 0 point followed by repeating digits 1 2 equals 0.121212 period period period

This can be written as the repeating decimal 0 point followed by repeating digits 1 2 equals 12 over 100 plus the fraction with numerator 12 and denominator 10 comma 000 plus period period period

Therefore a. sub 1 equals 12 over 100 and r equals 1 over 100.

The sum to infinity S sub infinity equals the fraction with numerator a. sub 1 and denominator 1 minus r becomes S sub infinity equals the fraction with numerator 12 over 100 and denominator 1 minus 1 over 100.

This simplifies to S sub infinity equals the fraction with numerator open paren 12 over 100 close paren and denominator open paren 99 over 100 close paren and S sub infinity equals 12 over 100 divided by 99 over 100 is found to be S sub infinity equals 12 over 99.

Sum to Infinity Proof

The proof of the sum to infinity formula is derived from the formula for the first n terms of a geometric series: Sn=a[1-rn]/[1-r]. If -1<r<1 then as n→∞, rn→0. Substituting rn with 0, the sum to infinity S∞=a[1-0]/[1-r], which simplifies to S∞=a/[1-r].

Here is the derivation of the sum to infinity of a geometric series in steps.

  1. The formula for the sum of the first n terms of a geometric series is S sub n equals the fraction with numerator a. times open paren 1 minus r to the n-th power close paren and denominator open paren 1 minus r close paren.
  2. Since negative 1 is less than r is less than 1, as n long right arrow infinity, r to the n-th power long right arrow 0.
  3. Substituting n equals infinity and r to the n-th power equals 0 into S sub n equals the fraction with numerator a. times open paren 1 minus r to the n-th power close paren and denominator open paren 1 minus r close paren becomes S sub infinity equals the fraction with numerator a. times open paren 1 minus 0 close paren and denominator open paren 1 minus r close paren.
  4. Simplifying, S sub infinity equals the fraction with numerator a. and denominator 1 minus r.
proof of the sum to infinity formula

This derivation works since the common ratio is defined to be between -1 and 1.

When a number between -1 and 1 is raised to a high power, its size decreases.

For example:

0.5 squared equals 0.25

0.5 cubed equals 0.125

0.5 to the fourth power equals 0.0625

As the power tends to an infinitely large value, the decimal size tends toward zero.

As such as n long right arrow infinity, r to the n-th power long right arrow 0.

This is why the ratio must be defined as -1<r<1 for a geometric series to have a sum to infinity.