Find the limit of sequence

aaaa202 · Jun 14, 2012

1. The problem statement, all variables and given/known data
I want to find the limit of ƩK(n+m,n)zⁿ
K(a,b) being the binomial coefficient.

2. Relevant equations
Cauchy root test?

3. 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 · Jun 14, 2012

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
[tex]\sum \begin{pmatrix}n+m \\ n\end{pmatrix}z^n[/tex]

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

aaaa202 · Jun 14, 2012

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 · Jun 14, 2012

aaaa202 said: ↑

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:
[tex] (-1)^n {n+m \choose n} = {-m \choose n}.[/tex] That should allow you to evaluate the sum explicitly.

RGV

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Find the limit of sequence

aaaa202

HallsofIvy

aaaa202

Ray Vickson

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