A convergent series for the Trigamma function $\psi_1(n) =\sum_{k=n}^{\infty} \frac1{k^2} $ I just came up with
the following
convergent series
for the Trigamma function
defined by
$\psi_1(n)
=\sum_{k=n}^{\infty} \frac1{k^2}
$.
\begin{align*}
\psi_1(n)
&=\lim_{m \to \infty} \sum_{j=1}^m \frac{(j-1)!}{j\prod_{i=0}^{j-1}(n+i)}\\
&=\frac1{n}+\frac{1}{2n(n+1)}+\frac{2}{3n(n+1)(n+2)}+\ldots\\
\end{align*}
This contrasts with
the usual asymptotic series
for $\psi_1(n)$
which is asymptotic,
does not converge,
and involves the
Bernoulli numbers.
I'm sure this isn't new,
but I could not find it here.
So,
my questions are,
as is often the case,
(1) Is this new?
(2) Is there a reasonably simple proof?
(Mine is moderately messy.)
I'll post my proof
in a few days
if anyone is interested.
Thanks
 A: Observe that $\prod_{i=0}^{j-1}(n+i)=\Gamma(n+j)/\Gamma(n)$, so that, as claimed,
\begin{align*}
\sum_{j=1}^\infty\frac{(j-1)!}{j\prod_{i=0}^{j-1}(n+i)}
&=\sum_{j=1}^\infty\frac1j\mathrm{B}(n,j)\\
&=\sum_{j=1}^\infty\frac1j\int_0^1 t^{n-1}(1-t)^{j-1}\,dt\\
&=\int_0^1 t^{n-1}\frac{-\log t}{1-t}\,dt\\
&=\int_0^1 t^{n-1}(-\log t)\sum_{j=0}^\infty t^j\,dt\\
&=\sum_{k=n}^\infty\int_0^1 t^{k-1}(-\log t)\,dt=\sum_{k=n}^\infty\frac1{k^2}.
\end{align*}
A: This is not a proof, not an answer but it is too long for comments.
Let
$$a_j=\frac{(j-1)!}{j\prod_{i=0}^{j-1}(n+i)}=\frac{(j-1)!}{j n (n+1)_{j-1}}\qquad \text{and} \qquad S_m=\sum_{j=1}^m a_j$$
$$\frac{(m+1) \Gamma (n+1) \Gamma (m+n+1)}{\Gamma (n) }S_m=$$
$$(m+1) n \psi ^{(1)}(n) \Gamma (m+n+1)-$$ $$\Gamma (m+1) \Gamma (n+1) \,
   _3F_2(1,m+1,m+1;m+2,m+n+1;1)$$
Simplifying
$$S_m=\psi ^{(1)}(n)-\frac{\Gamma (m+1) \Gamma (n)}{(m+1)
   \Gamma (m+n+1)} \, _3F_2(1,m+1,m+1;m+2,m+n+1;1)$$
What seems interesting is to make a contour plot of the second term; it shows how small it is.
We can also notice that for large values of $m$ $(m\gg n)$, the coefficient
$$\frac{\Gamma (m+1) \Gamma (n)}{(m+1) \Gamma (m+n+1)}\sim \frac{ \Gamma (n) }{m^{n+1}}$$
Unfortunately, I have not been to find the asymptotics of the hypergeometric function.
