I want to solve for the $$\sum_{b=0}^{\infty} \binom{a+b}{b}e^{0.3b}(0.3)^{a+b} $$ Can anyone give me idea how to start this? I'm thinking of manipulating it to look like the binomial series $$\sum_{j=0}^{n} \binom{n}{j}a^{j}b^{n-j} $$ but I have no idea regarding the changing of their upper limits. Any help would be greatly appreciated ☺


We can apply the binomial series expansion \begin{align*} \sum_{b=0}^\infty\binom{\alpha}{b}x^b=(1+x)^\alpha\qquad\qquad |x|<1, \alpha\in\mathbb{C}\tag{1} \end{align*}

We obtain \begin{align*} \color{blue}{\sum_{b=0}^\infty\binom{a+b}{b}e^{0.3b}(0.3)^{a+b}} &=0.3^a\sum_{b=0}^\infty\binom{-a-1}{b}\left(-0.3e^{0.3}\right)^b\tag{2}\\ &\color{blue}{=\frac{0.3^a}{\left(1-0.3e^{0.3}\right)^{a+1}}}\tag{3} \end{align*}


  • In (2) we use the binomial identity $\binom{-p}{q}=\binom{p+q-1}{q}(-1)^q$.

  • In (3) we apply the binomial series expansion according to (1) noting that $|-0.3e^{0.3}|<1$.


You can write:

$$\sum_{b=0}^{\infty}\binom{a+b}{b}\exp(0.3 b)0.3^{a+b} = (0.3)^a\sum_{b=0}^{\infty}\binom{a+b}{b}x^b $$

where $x = 0.3\exp(0.3)$ We can then compute this summation by noting that:

$$\binom{a+b}{b} = \frac{1}{a!}(b+a)(b+a-1)\cdots(b+1)$$

Then to compute the desired summation we can take the geometric series

$$\sum_{b=0}^{\infty}x^{a+b}= \frac{x^a}{1-x}$$

and differentiate both sides $a$ times.


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