A geometric series has the form S = a + ar + ar^2 + ar^3…+ ar^n. Here are some examples:
1 + 2 + 4 + 8…
1 + 1/2 + 1/4 + 1/8…
e + e^2 + e^3 + e^4…
It’s pretty easy to show that the partial sum of this kind of series is (a – ar^(n+1))/(1-r).
1. Multiply by r:
Sr = ar + ar^2 + ar^3 + ar^4…ar^(n+1)
2. Then subtract this from the original equation for S:
S – Sr
= [a + ar + ar^2 + ar^3…+ ar^n]
– [ar + ar^2 + ar^3 + ar^4…ar^(n+1)]
A bunch of stuff cancels, yielding
S – Sr = a – ar^(n+1)
3. Solve for S:
S(1 – r) = a – ar^(n+1)
S = (a – ar^(n+1))/(1-r)
That’s a remarkably simple answer. I bet Gauss came up with this proof first, it’s just the sort of thing he would do. Anyway, I did promise a mnemonic device.
Whether you know the trick or not, this proof’s easy to MSS (pronounced “miss”)
1. Multiply by r
3. Solve for S
See, you’d probably “miss” it if you didn’t know it already, but if you did know it you’d MSS it. Either way, it’s easy to miss/MSS. So clever!
In related news, I’m a dork. (Also in related news, it appears I’m going to be relearning LaTeX. *Sigh*)