Consider the series:

f(y)	= Σnyn-1     for |y| < 1 and n going from 1 to ∞
	= 1 + 2y + 3y2 + 4y3 + ...
	= d/dy (y + y2 + y3 + y4 + ...)
	= d/dy {y /(1 - y)}     putting 1/(1-y) = 1 + y + y2 + ...
	= 1/(1-y)2

In particular,
Σn/2ⁿ = 1/2 + 2 /2² + 3 /2³ + 4 /2⁴ = (1/2) f(1/2) = (1/2)/(1/2)² = 2
and
Σn/2^n/2 = (1/2^1/2) f(1/2^1/2) = (1/2^1/2) / ( 1 - 1/2^1/2 )² = 4 + 3 (2^1/2)

So, is this ritual useful? Would series such as these ever show up in a real-world problem?

Here's something from the brilliant physicist, Richard Feynman:

See: Math Stuff