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Economics · Ch 8 — Money and Banking

Appendix 3.1: The Sum of an Infinite Geometric Series

Appendix 3.1: The Sum of an Infinite Geometric Series

Appendix 3.1: The Sum of an Infinite Geometric Series

This appendix derives the formula for the sum of an infinite geometric series — the result the chapter uses to obtain the value of the money multiplier.

We want to find the sum of an infinite geometric series of the form

S=a+a r+a r2+a r3+⋯+a rn+⋯+∞S = a + a\,r + a\,r^2 + a\,r^3 + \cdots + a\,r^n + \cdots + \infty

where aa and rr are real numbers and 0<r<10 < r < 1. To compute the sum, multiply the equation by rr to obtain

r S=a r+a r2+a r3+⋯+a r n+1+⋯+∞r\,S = a\,r + a\,r^2 + a\,r^3 + \cdots + a\,r^{\,n+1} + \cdots + \infty

Subtracting the second equation from the first gives

S−r S=aS - r\,S = a

or,(1−r) S=a\text{or,}\quad (1 - r)\,S = a

which yields

S=a1−rS = \frac{a}{1 - r}

In the example used for the derivation of the money multiplier, a=1a = 1 and r=0.4r = 0.4. Hence the value of the infinite series is

11−0.4=53\frac{1}{1 - 0.4} = \frac{5}{3} …

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