Question

what will be the closed formula of the series from k = 0 to n of kr^k?

Answer 1

\[\sum_{k=0}^{\infty}kr^k\]?

Answer 2

think the gimmick is this looks like the derivative of
\[r^k\] but you are off by one in the exponent

Answer 3

yes

Answer 4

so first write
\[\sum_{k=0}^{\infty}r^k=\frac{1}{1-r}\]

Answer 5

then take the derivative to get
\[\sum kr^{k-1}=\frac{1}{(1-r)^2}\]

Answer 6

then multiply by r to get
\[\sum kr^k =\frac{r}{(1-r)^2}\]

Answer 7

as i recall that is the gimmick used here

Answer 8

you can basically ignore the problem about starting at k =0 or k = 1 because in this case k = 0 gives 0, so don't worry about that part.

Answer 9

why did we writ it as 1/(1-r)?

Answer 10

you mean
\[\sum r^k=\frac{1}{1-r}\]?

Answer 11

geometric series

Answer 12

i was assuming that this was well known

Answer 13

what alchemista said

Answer 14

http://en.wikipedia.org/wiki/Geometric_series

Answer 15

if we have to prove this then we have lots of work to do, but the question led me to believe that this was basically taken for granted

Answer 16

The proof of the closed form for a geometric series is in that wiki article.

Answer 17

Thanks y'all!