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:

• • a?is the first term
  • r?is 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 by?r
  • Subtract the second sum from the first
    • All the terms except the two should cancel out
  • Factorise and rearrange to make?S?the 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