A convergent series for the Trigamma function $\psi_1(n) =\sum_{k=n}^{\infty} \frac1{k^2} $

Question

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

Answer 1

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*}