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Sum to Infinity of a Geometric Series

Maths: Edexcel A Level Pure Maths, topic 4.2.9

Sum to Infinity of a Geometric Series

We can often find the sum of the terms of an infinite geometric sequence rather than the sum of only the first nn terms.

Infinite series

Infinite series

  • An infinite series is the sum of the terms of an infinite sequence.
  • For example, 2+4+8+16+...2+4+8+16+... is an infinite series, as is indicated by the "+...+...".
  • In sigma notation, ∑k=1∞2k\sum_{k=1}^{\infty}2^k is an infinite series as the upper limit of kk is infinity.
Divergent

Divergent

  • If the terms of a geometric series increase, then as the number of terms in the series increases to infinity, the value of the sum will also increase to infinity.
  • We say that this series is divergent.
  • The value of the sum diverges away from being a fixed, known value.
  • All non-zero arithmetic series are divergent.
Convergent

Convergent

  • If the terms of a geometric series decrease, then as the number of terms in the series increases to infinity, the value of the sum gets closer and closer to a fixed value.
  • We say that this series is convergent.
  • The value of the infinite sum has a fixed value that we can find.
  • The more finite terms we have in the series, the closer the sum will be to that fixed value.
Common ratio

Common ratio

  • We can determine if a sequence is convergent by looking at the common ratio of the sequence, rr:
  • A sequence is convergent if ∣r∣<1|r|<1.
  • We can also write this as −1<r<1-1<r<1.
Sum to infinity

Sum to infinity

  • The sum of the first nn terms of a geometric series is Sn=a(1−rn)1−rS_n = \frac{\small a(1-r^n)}{\small 1-r}.
  • If ∣r∣<1|r|<1, then as nn gets larger, the term rnr^n gets smaller.
  • This means for n=∞n=\infty, rn=0r^n=0.
  • The sum to infinity of a geometric series is equal to:
  • S∞=a1−rS_{\infty} = \frac{a}{1-r}
Example

Example

  • Determine whether the infinite geometric series with a=27a=27 and r=13r=\frac{1}{3} is a divergent or convergent series.
  • Find the sum of the infinite series if it is convergent.
Common ratio

Common ratio

  • If the infinite series is convergent, then ∣r∣<1|r|<1.
  • Substituting in r=13r=\frac{1}{3}, we can see that:
  • ∣13∣<1\left|\frac{1}{3}\right|<1
  • So the infinite series is convergent and we can find the sum.
Find the sum

Find the sum

  • The sum of a convergent infinite series is given by:
  • S∞=a1−rS_{\infty}=\frac{\small a}{\small 1-r}
  • Substituting in a=27a = 27 and r=13r=\frac{1}{3}, we get:
  • S∞=271−13=2723=40.5S_{\infty}=\frac{27}{\small 1-\frac{1}{3}}=\frac{27}{\small \frac{2}{3}} = 40.5

Test yourself on Sum to Infinity of a Geometric Series

  • Which of the following series is convergent?

  • Which of the following geometric series is convergent?

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