S_n = x + 2x2 + 3x3 + ... + nxn
xS_n = x2 + 2x3 + ... + (n-1)xn + nxn+1
(1-x)S_n = [x + x2 + x3 + ... + xn ] - nxn+1 [subtracting]
(1-x)S_n = x(1 - xn+1)/(1-x) - nxn+1 [sum of geometric series]
=> S_n = x(1 - xn+1)/(1-x)^2 - nxn+1/(1-x) [for x not = 1]
= x(nxn+1 - (n+1)xn + 1)/(x-1)^2
BTW The series in the question is formally known as an arithmetic-geometric series.
blackjack
<a href="http://c.casalemedia.com/c?s=51816&f=3&id=0" target="_blank"><img src="http://as.casalemedia.com/s?s=51816&u=http://perplexus.info&f=3&id=0&if=0" width="120" height="600" border="0"></a>
flooble's webmaster puzzle
Let the record show- another non-calculus variation
