Chaitanya's Random Pages

November 27, 2011

A few cute mathematical series

Filed under: mathematics — ckrao @ 3:11 am

From the geometric series \sum_{k=0}^{\infty} z^k = \frac{1}{1-z} we can arrive at a couple of attractive-looking series:

\displaystyle \sum_{n = 0}^{\infty} \frac{a^n}{(a+1)^{n+1}} = 1 \quad \quad (1)

\displaystyle \sum_{n=0}^{\infty} \left(1 - \frac{1}{a}\right)^n = a \quad \quad (2)

For example, \sum_{n=0}^{\infty} \frac{3^n}{4^{n+1}} = \sum_{n=0}^{\infty} \frac{4^n}{5^{n+1}} = 1 and \sum_{n=0}^{\infty} \left(\frac{7}{8}\right)^n = 8.

Furthermore, from the sums \sum_{k=1}^{\infty}k z^k = \frac{z}{(1-z)^2} and  \sum_{k=1}^{\infty} k^2z^k = \frac{z(1+z)}{(1-z)^3} we obtain (after setting z = a/(a+1))

\displaystyle \sum_{n = 1}^{\infty} \frac{na^{n-1}}{(a+1)^{n+1}} = 1 \quad \quad (3)

\displaystyle \sum_{n = 1}^{\infty} \frac{n^2a^n}{(a+1)^n} = a(a+1)(2a+1) \quad \quad (4)

This last sum reminds me of the identity \sum_{n = 1}^a n^2 = a(a+1)(2a+1)/6 for positive integers a, though a appear in the sums in different places!

So for what values of a are sums (1)-(4) valid (i.e. when do they converge)? For any positive integer k the sum \sum_{n=1}^{\infty} n^k z^n converges when |z| < 1, so this means sums (1), (3) and (4) converge when |a/(a+1)| < 1. In other words, |a| < |a+1|, or equivalently, a is closer to 0 than to -1. Hence the real part of a must be at least -1/2. Similarly, sum (2) converges when |(a-1)/a| < 1, which is equivalent to the real part of a being at least 1/2.

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