## Of staircases and textbooks

October 27, 2009

This week, I’m talking you about a little identity that crops up often in the study of algorithms and which isn’t found in formula compendia—anyway, none that I have. I’m talking about this function:

$\displaystyle C_{n,a}=\sum_{i=1}^n i a^i$

This is a variation on the Gabriel’s Staircase function that does not have an infinite number of terms. Let us solve it without supposing that $0.