Tackling the divergence or convergence of a binomial sum

Question

For the sums of the type, $$\sum_x \binom{x+c}{c}$$ where $c$ is a constant integer, are there known (simple) methods or line of arguments for discussing whether such sums are divergent or not? Specially, when one takes the limit of the sum going to infinity (so $x:0\to \infty$). Intuitively, I would imagine it must be diverging, because with increasing $x$ we are going to have ever larger positive numbers popping up in the sum. But at a rigorous level, I have no idea how to approach this.

Answer 1

The sum of an increasing positive sequence diverges. Indeed, if $u_{n+1}>u_n>0$ for all $n$, then $u_n>u_0>0$ for all $n$, and $$ \sum_{n=0}^N u_n > (N+1)\, u_0\, . $$ Taking the limit as $N$ goes to infinity gives $\sum u_n = +\infty$. Going back to the original question where $u_n = \binom{n+c}{c}$, we can see that $u_0 = \binom{c}{c} = 1$ is positive. Moreover, \begin{aligned} u_{n+1} & = \frac{(n+c+1)!}{(n+1)!\, c!} \\ &= \frac{(n+1)\times\dots\times (n+c)}{ c!} \frac{n+c+1}{n+1} \\ &= u_n \left( 1 + \frac{c}{n+1} \right) \\ &> u_n \, . \end{aligned} Therefore, $(u_n)_{n\in \mathbb{N}}$ is an increasing positive sequence, and its sum diverges. Note that $u_n = \binom{n+c}{c}$ is the maximal number of coefficients of a $n$-variate polynomial with degree $c$ [1].