Mathematical Methods in the Physical Sciences · Problem 1.2

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · geometric-series

Problem

Derive the formula (1.4) for the sum

S_n

of the geometric progression

S_n = a + ar + ar^2 + \cdots + ar^{n-1}

. Hint: Start by multiplying

S_n

r

and subtracting the result from

S_n

. If necessary, you can look up the derivation in an algebra book. Show that the geometric series (1.6) converges if and only if

|r| < 1

; also show that if

|r| < 1

, the sum is given by equation (1.8).