# Does this infinite series make sense ?

1. Apr 8, 2008

### mhill

given the series

$$g(x)= \sum_{n=0}^{\infty}\frac{a_{n}}{\sqrt {x-n}}$$

where the coefficients a_n are real numbers my question is does the above makes sense ? i mean since we are summing over all positive integers , no matter how big we choose 'x' there will be a factor so x-n n=0,1,2,3,4,.................. (x-n) <0 then g(x) is complex no matter what real x we put.

If we approximate the series by an integral $$\int_{0}^{\infty}dr a(r) (x-r)^{-1/2}$$ we get a similar result, i do not know if this is paradoxical and i should have taken the absolute value |x| inside the square root.

2. Apr 8, 2008

### mathman

It makes sense mathematically. You end up with a complex function. Convergence may be another matter - depending on the behavior of an.