# Sum or upped bound of geometrico-harmonic series

1. Jun 20, 2010

### aamir.ahmed

Hi,
I need help to determine the upper bound of this infinite series.
$$\sum_{k=p+1}^{\infty} \frac{1}{k} a^k \ \ \ \ ; a \leq 1$$

The paper I am reading reports the upper bound to be,
$$\sum_{k=p+1}^{\infty} \frac{1}{k} a^k \leq \frac{1}{p+1}\sum_{k=p+1}^{\infty} a^k = \frac{1}{p+1} \cdot \frac{a^{p+1}}{1-a}$$

I totally cannot understand how could he factor 1/k coefficient out of the summation and replace it with 1/(p+1). Please point me in the right direction.

Last edited: Jun 20, 2010
2. Jun 20, 2010

### CRGreathouse

k is at least p+1, so 1/k is at most 1/(p+1).