# Math Help - combinatorial sum

1. ## combinatorial sum

$\sum_{n=2}^x {1\over{x}} (1+{1\over{x}})^{n-2}$

anyone have a way of making this prettier?

2. Originally Posted by jbpellerin
$\sum_{n=2}^x {1\over{x}} (1+{1\over{x}})^{n-2}$

anyone have a way of making this prettier?
$\sum_{n=2}^x {1\over{x}} (1+{1\over{x}})^{n-2}$ is equivalent to

$\sum_{n=2}^x \frac {1}{x}(\frac{x+1}{x})^{n-2}$

$\sum_{n=2}^x \frac{(x+1)^{n-2}}{{x}^{n-1}}$

Can you take it from there?

3. actually it turned out to be pretty easy
$(1+{1\over{x}})^{x-1} -1$
it's a geometric series

4. First observation : you can factor out 1/x from the sum :
$=\frac 1x \sum_{n=2}^x \left(1+\frac 1x\right)^{n-2}$

Second observation : what you said > geometric series !