Skip to main content

[Community Question] Algebra-Precalculus: series $\sum_{n=1}^\infty \sum_{m=1}^\infty \frac{(m+n-1)!}{m!(n-1)!n!(m-1)!}a^m b^n.$

One of our user asked:

Can anyone please help me with the computation of following series:

$$\sum_{n=1}^\infty \sum_{m=1}^\infty \frac{(m+n-1)!}{m!(n-1)!n!(m-1)!}a^m b^n.$$

My thoughts: Since $$\displaystyle \frac{(m+n-1)!}{m!(n-1)!n!(m-1)!} = \frac{\binom{m+n-1}{m}\binom{m+n-1}{m-1}}{(m+n-1)!},$$ by some arrangement this may be the probability of a hypergeometric distribution.

Any help would be appreciated, thanks!


Labels:

Comments

Post a Comment

Popular posts from this blog

[Community Question] Calculus: Manifold with boundary - finding the boundary

One of our user asked: I have the manifold with boundary $M:= \lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1\geq 0, x_1^2+x_2^2+x_3^2=1\rbrace \cup\lbrace (x_1,x_2,x_3) \in \mathbb R^3 : x_1= 0, x_1^2+x_2^2+x_3^2\leq1\rbrace$ and I need to find the boundary of this manifold. I think it is $\lbrace (x_1,x_2,x_3) \in \mathbb R^n : x_1= 0, x_2^2+x_3^2=1\rbrace$ , the other option is that the boundary is the empty set? I think the first is right? Am I wrong?
Post a Comment
Read more

[Community Question] Linear-algebra: Are linear transformations between infinite dimensional vector spaces always differentiable?

One of our user asked: In class we saw that every linear transformation is differentiable (since there's always a linear approximation for them) and we also saw that a differentiable function must be continuous, so it must be true that all linear operators are continuous, however, I just read that between infinite dimensional vector spaces this is not necessarily true. I would like to know where's the flaw in my reasoning (I suspect that linear transformations between infinite dimensional vector spaces are not always differentiable).
Post a Comment
Read more

Order of elements of the Prüfer groups $\mathbb{Z}(p^{\infty})$

Let $\mathbb{Z}(p^{\infty})$ be defined by $\mathbb{Z}(p^{\infty}) = \{ \overline{a/b} \in \mathbb{Q}/ \mathbb{Z} / a,b \in \mathbb{Z}, b=p^i$ $ with$ $ i \in \mathbb{N} \}$ , I wish show that any element in $\mathbb{Z}(p^{\infty})$ has order $p^n$ with $n \in \mathbb{N}$ . i try several ways but I have not been successful, some help ?? thank you
Post a Comment
Read more