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A problem on my (last week's) real analysis homework boiled down to proving that, for $a>-1$, $$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{1}{(a+k)^2}.$$ Mathematica confirms this is true, but I couldn't even prove the convergence of the original series (the one on the left), much less demonstrate that it equaled this other sum; the ratio test is inconclusive, and the root test and others seem hopeless. It was (and is) quite a frustrating problem. Can someone explain how to go about tackling this?

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You can obtain convergence using Raabe's Ratio Test. –  xen Oct 25 '11 at 8:51
    
Would going the hypergeometric route be a nuke for you here? –  J. M. Oct 25 '11 at 9:05
    
@J.M. I'm only vaguely familiar with hypergeometric series, but certainly I'd be interested in whatever approach you have; however, this occurred in a problem in Chapter 2 of Folland's Real Analysis, and I don't believe that we are assumed to have seen hypergeometric series before, so I would hope there is some relatively straightforward way of proving it. –  Zev Chonoles Oct 25 '11 at 9:11
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@Hans: methinks the difference calculus route could be pedagogical; I say go for it! –  J. M. Oct 25 '11 at 10:21
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This was rather a fun question to watch evolving; thanks to everyone! The only thing missing now is a combinatorial proof using objects counted by generating functions given by the summands on the two sides :-) –  joriki Oct 25 '11 at 10:46
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up vote 19 down vote accepted

This uses a reliable trick with the Beta function. I say reliable because you can use the beta function and switching of the integral and sum to solve many series very quickly.

First notice that $$\prod_{i=1}^{n}(a+i)=\frac{\Gamma(n+a+1)}{\Gamma(a+1)}.$$ Then

$$\frac{(n-1)!}{\prod_{i=1}^{n}(a+i)}=\frac{\Gamma(n)\Gamma(a+1)}{\Gamma(n+a+1)}=\text{B}(n,a+1)=\int_{0}^{1}(1-x)^{n-1}x{}^{a}dx.$$ Hence, upon switching the order we have that $$\sum_{n=1}^{\infty}\frac{(n-1)!}{n\prod_{i=1}^{n}(a+i)}=\int_{0}^{1}x^{a}\left(\sum_{n=1}^{\infty}\frac{(1-x)^{n-1}}{n}\right)dx.$$ Recognizing the power series, this is $$\int_{0}^{1}x^{a}\frac{-\log x}{1-x}dx.$$ Now, expand the power series for $\frac{1}{1-x}$ to get $$\sum_{m=0}^{\infty}-\int_{0}^{1}x^{a+m}\log xdx.$$ It is not difficult to see that $$-\int_{0}^{1}x^{a+m}\log xdx=\frac{1}{(a+m+1)^{2}},$$ so we conclude that $$\sum_{n=1}^{\infty}\frac{(n-1)!}{n\prod_{i=1}^{n}(a+i)}=\sum_{m=1}^{\infty}\frac{1}{(a+m)^{2}}.$$

Hope that helps,

Remark: To evaluate the earlier integral, notice that $$-\int_{0}^{1}x^{r}\log xdx=\int_{1}^{\infty}x^{-(r+2)}\log xdx=\int_{0}^{\infty}e^{-u(r+1)}udu=\frac{1}{(r+1)^{2}}\int_{0}^{\infty}e^{-u}udu. $$ Alternatively, as Joriki pointed out, you can just use integration by parts.

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+1, very nice. You can also get that last integral by integration by parts. –  joriki Oct 25 '11 at 9:26
    
@Joriki: Hmmm that is a good point! Certainly the easier way. (I was in the Gamma Function mindset) –  Eric Naslund Oct 25 '11 at 9:27
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I see my trouble now; the original problem was actually to show that $$\int_0^1x^a(1-x)^{-1}\log(x)\,dx=\sum_{k=1}^\infty\frac{1}{(k+a)^2}$$ with the instruction to expand as an infinite series and justify switching the order; but I expanded the logarithm, and ended up at the series $$\sum_{n=1}^{\infty}\frac{(n-1)!}{n\prod_{i=1}^{n}(a+i)}$$ with nowhere to go :) Thanks for your excellent answer! –  Zev Chonoles Oct 25 '11 at 9:36
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@Eric: Induction on $n$; for $n=0$ we know it's $\frac{1}{a+1}$, then using integration by parts I got $$\int_0^1x^a(x-1)^{k}\,dx=\left.\frac{x^{a+1}(x-1)^{k}}{a+1}\right|_0^1-\frac{k‌​}{a+1}\int_0^1x^{a+1}(x-1)^{k-1}\,dx=$$ $$(0-0)-\frac{k}{a+1}\left(\int_0^1x^{a+1}(x-1)^{k-1}\,dx\right)$$ which shows that $$\int_0^1x^a(x-1)^{n}\,dx=\frac{(-1)^{n}n!}{\prod_{i=1}^{n+1}(a+i)}$$ –  Zev Chonoles Oct 25 '11 at 9:43
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@Zev: oooo right, because $k$ is still an integer! Ok thanks, that is very good! You might also like the proof on Wikipedia for the Beta function/ Gamma function identity. It is basically all the one needs to know about the beta function: (cool integration tricks in the plane) en.wikipedia.org/wiki/… –  Eric Naslund Oct 25 '11 at 9:47
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Since at least J. M. asked for it, here's another solution for the case when $a$ is a natural number.

I'll use the forward difference operator $\Delta$, defined by $\Delta f(n) = f(n+1) - f(n)$, and the falling factorial defined by $$ n^{\underline{a}} = \begin{cases} n(n-1)(n-2) \dots (n-a+1), & a > 0, \\ 1, & a=0 \\ \frac{1}{(n+1)(n+2) \dots (n+|a|)}, & a < 0, \end{cases} $$ and satisfying $\Delta n^{\underline{a}} = a n^{\underline{a-1}}$.

The summand, which I'll denote by $F_a(n)$, can be rewritten as $$ F_a(n) = \frac{(n-1)!}{n\prod_{i=1}^n(a+i)} = \frac{(n-1)! a!}{n (a+n)!} = \frac{a!}{n \cdot n(n+1)(n+2) \dots (n+a)} $$ $$= \frac{(a-1)!}{n} \left( -(-a) (n-1)^{\underline{-(a+1)}}\right) = -\frac{(a-1)!}{n} \Delta\left( (n-1)^{\underline{-a}}\right). $$ Using the rule $\Delta(f(n)g(n)) = \Delta f(n) \, g(n+1) + f(n) \Delta g(n)$, we get $$ F_a(n) = - \Delta\left( \frac{(a-1)!}{n} (n-1)^{\underline{-a}}\right) + \Delta\left( \frac{(a-1)!}{n} \right) \, n^{\underline{-a}} $$ $$ = - \Delta\left( \frac{(a-1)!}{n \cdot n (n+1) \dots (n+a-1)} \right) + (a-1)! \left( \frac{1}{n+1} - \frac{1}{n} \right) \frac{1}{(n+1)\dots (n+a)} $$ $$= - \Delta\left( \frac{(a-1)!}{n \cdot n(n+1) \dots (n+a-1)} \right) + F_{a-1}(n+1) - (a-1)! \Delta\left( \frac{(n-1)^{\underline{-a}}}{-a} \right). $$

Summing over $n \ge 1$ gives (because of telescoping in the sums-of-deltas) $$ \sum_{n=1}^{\infty} F_a(n) = \frac{(a-1)!}{1 \cdot a!} + \sum_{n=1}^{\infty} F_{a-1}(n+1) - \frac{(a-1)!}{a} 0^{-\underline{a}} $$ $$ = \frac{1}{a} + \sum_{m=2}^{\infty} F_{a-1}(m) - \frac{1}{a^2} $$ $$ = \sum_{m=1}^{\infty} F_{a-1}(m) - \frac{1}{a^2} $$ (since $F_{a-1}(1) = 1/a$).

Finally, since $F_0(n) = 1/n^2$, we obtain after using this result to work our way down $n$ steps that $$ \sum_{n=1}^{\infty} F_a(n) = \sum_{n=1}^{\infty} F_{a-1}(n) - \frac{1}{a^2} = \dots = \sum_{n=1}^{\infty} F_0(n) - \left( \frac{1}{a^2} + \dots + \frac{1}{1^2} \right) = \sum_{n=a+1}^{\infty} \frac{1}{n^2} = \sum_{k=1}^{\infty} \frac{1}{(k+a)^2}. $$

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I was expecting you'd be using a as the base and then connect rising and falling factorials, but this is nice! –  J. M. Oct 26 '11 at 3:33
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