The sum that I need help integrating is as follows: $$\int^{k}_{1}\frac{1}{n}+\frac{1}{n(n+k)}+\frac{1}{n(n+k)(n+2k)}+\frac{1}{n(n+k)(n+2k)(n+3k)}+\ …$$ I was unable to find information on how to do a general integration of a rising factorial to obtain a closed form. If anyone could point me in the right direction I would greatly appreciate it.

I need help proving the following identity: $$\frac{(6n)!}{(3n)!} = 1728^n \left(\frac{1}{6}\right)_n \left(\frac{1}{2}\right)_n \left(\frac{5}{6}\right)_n.$$ Here, $$(a)_n = a(a + 1)(a + 2) \cdots (a + n – 1), \quad n > 1, \quad (a)_0 = 1,$$ is the Pochhammer symbol. I do not really know how one converts expressions involving factorials to products of the Pochhammer […]

How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ This is about as far as I got: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ $$\sum_{j=2}^\infty\frac{2^j j!}{(j)_j} = \frac{2^2 2!}{(2)_2} + \frac{2^3 3!}{(3)_3} + \frac{2^4 4!}{(4)_4}+\cdots \implies$$ $$?$$ where $(x)_n$ denotes the Pochhammer symbol. Maybe reduction isn’t the way to go?

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