4.4.2 Geometric Series | Edexcel A Level Maths: Pure Revision Notes 2018 (2024)

How do I find the sum of a geometric series?

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

• The following formulae will let you find the sum of the first n terms of a geometric series:

or

• • ais the first term
  • ris the common ratio

• The one on the left is more convenient if r<1, the one on the right is more convenient if r>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 byr
  • Subtract the second sum from the first
    • All the terms except two should cancel out
  • Factorise and rearrange to makeSthe subject

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