What Is the Limit of the Sequence Defined by the Sum of Binomial Coefficients?

[aaaa202]

Homework Statement

I want to find the limit of ƩK(n+m,n)zⁿ
K(a,b) being the binomial coefficient.

Homework Equations

Cauchy root test?

The Attempt at a Solution

Trying the cauchy root test I get:

1/R = lim_n->∞[(K(n+m,n))^½]

But what do I do from here?

 

[HallsofIvy]

The "Cauchy root test" tells you whether or not a series converges. It says nothing about what it converges to. If I read this correctly, you have
$$\sum \begin{pmatrix}n+m \\ n\end{pmatrix}z^n$$

The sum is over n with m fixed? And it is a finite sum? n goes from 0 to what?

 

[aaaa202]

well maybe I named it wrong, but I meant the formula stated above, which gives an explicit expression for the radius of convergence, R.
And the sum is from zero to infinity. Sorry for the lack of information :)

 

[Ray Vickson]

well maybe I named it wrong, but I meant the formula stated above, which gives an explicit expression for the radius of convergence, R.
And the sum is from zero to infinity. Sorry for the lack of information :)

If you set z = -t, the coefficient of t^n is the "negative binomial" coefficient:
$$(-1)^n {n+m \choose n} = {-m \choose n}.$$ That should allow you to evaluate the sum explicitly.

RGV