• Support PF! Buy your school textbooks, materials and every day products Here!

Find the limit of sequence

  • Thread starter aaaa202
  • Start date
  • #1
1,170
3

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 = limn->∞[(K(n+m,n))½]

But what do I do from here?
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,794
924
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?
 
  • #3
1,170
3
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 :)
 
  • #4
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728
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
 
Last edited:

Related Threads for: Find the limit of sequence

  • Last Post
Replies
1
Views
940
  • Last Post
Replies
3
Views
753
  • Last Post
Replies
16
Views
1K
Replies
4
Views
1K
  • Last Post
Replies
11
Views
4K
  • Last Post
Replies
1
Views
2K
Replies
7
Views
917
  • Last Post
Replies
8
Views
1K
Top