Geometric Series (OCR A Level Maths: Pure)

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Geometric Series

How do I find the sum of a geometric series?

  • A geometric series is the sum of the terms of a geometric sequence

Geom Series Illustr, A Level & AS Level Pure Maths Revision Notes 

  • The following formulae will let you find the sum of the first n terms of a geometric series:
S subscript n equals fraction numerator a left parenthesis 1 minus r to the power of n right parenthesis over denominator 1 minus r end fraction   or   S subscript n equals fraction numerator a left parenthesis r to the power of n minus 1 right parenthesis over denominator r minus 1 end fraction
    • is the first term
    • is the common ratio

  • The one on the left is more convenient if < 1, the one on the right is more convenient if > 1
  • The a and the r in those formulae are exactly the same as the ones used with geometric sequences

How do I prove the geometric series formula?

  • Learn this proof of the geometric series formula – you can be asked to give it in the exam:
    • Write out the sum once
    • Write out the sum again but multiply each term by r
    • Subtract the second sum from the first
      • All the terms except two should cancel out
    • Factorise and rearrange to make the subject

Geom Series Proof, A Level & AS Level Pure Maths Revision Notes 

What is the sum to infinity of a geometric series?

  • If (and only if!) |r| < 1, then the geometric series converges to a finite value given by the formula
S subscript infinity equals fraction numerator a over denominator 1 minus r end fraction

  •  S is known as the sum to infinity
  • If |r| ≥ 1 the geometric series is divergent and the sum to infinity does not exist

Examiner Tip

  • The geometric series formulae are in the formulae booklet – you don't need to memorise them
  • You will sometimes need to use logarithms to answer geometric series questions (see Exponential Equations)

Worked example

Geom Series Example, A Level & AS Level Pure Maths Revision Notes

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Roger

Author: Roger

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.