Infinite geometric series

Objective. Establish convergence and calculate limiting sums.

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Learn: the key idea

An infinite geometric series sums infinitely many terms of a geometric sequence, converging to a finite limit only when the common ratio satisfies |r|<1, using the formula S∞=a/(1−r). Example: the series 8+4+2+1+... has a=8, r=1/2, so S∞=8/(1−0.5)=16, meaning the partial sums get arbitrarily close to 16 without exceeding it. If |r|≥1, the series diverges and no finite sum exists.

Worked example

Find the sum to infinity of the geometric series 12+6+3+1.5+...

1 $Identify a=12 and common ratio r=6/12=1/2$
2 $Confirm |r|<1 so the infinite series converges$
3 $Apply the formula S\infty=a/(1-r)=12/(1-0.5)$
4 $Simplify: 12/0.5=24$

Answer: 24

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Common mistake. Students apply S∞=a/(1−r) even when |r|≥1, producing a meaningless finite answer for a divergent series; always verify |r|<1 before using the infinite sum formula.

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