GP and geometric series

A surprising fact: if the common ratio has absolute value less than $1$, adding up infinitely many terms of a geometric progression still gives a finite result.

Observation — GP and geometric series

For $|q|<1$ the GP even has a finite infinite sum: $\sum_{k=0}^{\infty}a_1\,q^k = \lim_{n\to\infty}a_1\,\frac{1-q^n}{1-q} = \frac{a_1}{1-q}.$ This is the geometric series, already met in Year Two with the computation of the area under an infinite staircase of rectangles and taken up formally in the last section of this chapter.

The mechanism is that of the limit: the partial sums $a_1\,\frac{1-q^n}{1-q}$ form a sequence, and for $|q|<1$ the term $q^n$ tends to $0$, leaving the limiting value $\frac{a_1}{1-q}$.

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Topics: Sequences
Concepts: Limit of a sequence · Geometric progression · Geometric series