Prove the limit exists

I was toying around with Abel summability when I stumbled upon a limit I could not prove existed.

$$\lim_{x\to-1^+}\sum_{n=2}^\infty x^n\ln(n)\tag{*}$$

While it may not be clear that such a limit could converge, it may be helpful to note a similar example:

$$\lim_{x\to-1^+}\sum_{n=1}^\infty nx^{n-1}=\lim_{x\to-1^+}\frac1{(1-x)^2}=\frac14$$

However, a lack of closed form of $(*)$ makes it difficult for me to show it converges. WolframAlpha returns the series as a derivative of the Lerchphi function, though it doesn’t seem quite helpful.

By considering

$$f_k(x)=\sum_{n=2}^k x^n\ln(n)$$

I find that

$$f_{35}(-0.75)-0.25f’_{35}(-0.75)=0.225803586648$$

Which is a quick linear approximation of $f_{35}(x)$ centered at $x=-\frac34$. This agrees with what I think to be the limit:

$$\lim_{x\to-1^+}\sum_{n=2}^\infty x^n\ln(n)\stackrel?=\eta'(0)=\frac12\ln\left(\frac\pi2\right)=0.225791352645\tag{**}$$

Where $\eta(s)$ is the Dirichlet eta function.

I also tried considering more elementary approaches to showing the limit exists, such as using $\ln(n+1)=\ln(n)+\mathcal O(n^{-1})$, however, I could not make use of it.

Bonus points if you can prove $(**)$.

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Recall the archetypal Frullani integral

$$\int_{0}^{\infty} \frac{e^{-u} – e^{-nu}}{u} \, du = \log n.$$

Since the integrand is non-negative for all $n \geq 1$, when $x \in [0, 1)$ we can apply the Tonelli’s theorem to interchange the sum and integral unconditionally to get

\begin{align*}
\sum_{n=1}^{\infty} x^n \log n
&= \int_{0}^{\infty} \left( \sum_{n=1}^{\infty} x^n \cdot \frac{e^{-u} – e^{-nu}}{u} \right) \, du \\
&= \frac{x^2}{1-x} \int_{0}^{\infty} \frac{e^{-u}(1 – e^{-u})}{u(1-xe^{-u})} \, du. \tag{1}
\end{align*}

Notice that the last integral converges absolutely. So this computation can be fed back to the Fubini’s theorem, showing that exactly the same computation can be carried out to prove $\text{(1)}$ for all $|x| < 1$.

Now we would like to take limit as $x \to -1^{+}$. When $x \in (-1, 0]$, the integrand of the last integral of $\text{(1)}$ is uniformly bounded by the integrable function $u^{-1}e^{-u}(1-e^{-u})$. Therefore by the dominated convergence theorem, as $x \to -1^{+}$ we have

$$\lim_{x\to -1^+} \sum_{n=1}^{\infty} x^n \log n = \frac{1}{2} \int_{0}^{\infty} \frac{e^{-u}(1 – e^{-u})}{u(1+e^{-u})} \, du.$$

This already proves that the limit exists, but it even tells more that the limit is indeed $\eta'(0)$. To this end, we first perform integration by parts to remove the pesky factor $u$ in the denominator. Then the right-hand side becomes

$$\frac{1}{2} \int_{0}^{\infty} \frac{e^{-u}(1 – e^{-u})}{u(1+e^{-u})} \, du = \int_{0}^{\infty} \left( \frac{e^u}{(e^u + 1)^2} – \frac{e^{-u}}{2} \right) \log u \, du$$

In order to compute this integral, it suffices to prove the following claim.

Claim. We have

$$\int_{0}^{\infty} e^{-u} \log u = -\gamma, \qquad \int_{0}^{\infty} \frac{e^u \log u}{(e^u + 1)^2} \, du = -\frac{\gamma}{2} + \eta'(0).$$

Notice that the first claim is an immediate consequence of the identity $\psi(1) = -\gamma$, where $\psi$ is the digamma function. Next, term-wise integration gives

$$\int_{0}^{\infty} \frac{u^{s-1}}{e^{\alpha u} + 1} \, du = \alpha^{-s} \Gamma(s)\eta(s)$$

where $\alpha > 0$ and $s$ is initially assumed to satisfy $\Re(s) > 1$ (so that interchanging the summation and integration works smoothly). Differentiating both sides w.r.t. $\alpha$ gives

$$\int_{0}^{\infty} \frac{u^s e^{\alpha u}}{(e^{\alpha u} + 1)^2} \, du = \alpha^{-s-1} \Gamma(s+1)\eta(s).$$

Although we initially assumed $\Re(s) > 1$, now both sides define an analytic function for $\Re(s) > -1$, hence by the principle of analytic continuation this identity extends to this region as well. Now plugging $\alpha = 1$ and differentiating both sides w.r.t. $s$, we get

$$\int_{0}^{\infty} \frac{u^s e^u \log u}{(e^u + 1)^2} \, du = \Gamma(s+1)\psi(s+1)\eta(s) + \Gamma(s+1)\eta'(s).$$

Plugging $s = 0$ and using known values $\psi(1) = -\gamma$ and $\eta(0) = \frac{1}{2}$, this yields

$$\int_{0}^{\infty} \frac{e^u \log u}{(e^u + 1)^2} \, du = -\frac{\gamma}{2} + \eta'(0).$$

$\lim_{x\to-1^+}\sum_{n=2}^\infty x^n\ln(n)\tag{$*$}$

With Wolfy’s help,
I confirmed that
the limit is
$\frac12\ln(\pi/2) \approx 0.225791…$.

If we just plug in
$x = -1$,
this is

$\begin{array}\\ \sum_{n=2}^{2m+1} (-1)^n\ln(n) &=\sum_{n=1}^m (\ln(2n)-\ln(2n+1))\\ &=\sum_{n=1}^m \ln(\frac{2n}{2n+1})\\ &=\sum_{n=1}^m -\ln(\frac{2n+1}{2n})\\ &=-\sum_{n=1}^m \ln(1+\frac{1}{2n})\\ &=-\sum_{n=1}^m (\frac{1}{2n}+O(\frac{1}{n^2}))\\ &=-\frac12 \ln(m) + O(1)\\ \end{array}$

The sum to $2m+2$
would then be

$\begin{array}\\ -\frac12 \ln(m) + O(1)+\ln(2m+2) &=-\frac12 \ln(m)+\ln(2)+\ln(m+1) + O(1)\\ &=-\frac12 \ln(m)+\ln(2)+\ln(m)+\ln(1+1/m) + O(1)\\ &=\frac12 \ln(m)+O(1)\\ \end{array}$

Therefore the sum of
the even and odd sums
is $O(1)$;
with a little more work
I could get a more precise estimate
(by expanding
$\ln(1+\frac{1}{2n})$
it looks like the sum
would involve
$\gamma$ and
$\sum (-1)^k\zeta(k)/k$
).

So,
by my sloppy thinking,
the Cesaro sum converges
so the limit exists.

Here is a more accurate
version of the computation.

Lets look at
the partial sums.

$\begin{array}\\ \sum_{n=2}^{2m+1} (-1)^n\ln(n) &=\sum_{n=1}^m (\ln(2n)-\ln(2n+1))\\ &=\sum_{n=1}^m \ln(\frac{2n}{2n+1})\\ &=\sum_{n=1}^m -\ln(\frac{2n+1}{2n})\\ &=-\sum_{n=1}^m \ln(1+\frac{1}{2n})\\ &=-\sum_{n=1}^m \sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k(2n)^k})\\ &=-\sum_{k=1}^{\infty}\frac{(-1)^{k=1}}{k2^k}\sum_{n=1}^m \frac{1}{n^k})\\ &=-\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k2^k}\sum_{n=1}^m \frac{1}{n^k})\\ &=-\frac12\sum_{n=1}^m \frac{1}{n}-\sum_{k=2}^{\infty}\frac{(-1)^{k-1}}{k2^k}\sum_{n=1}^m \frac{1}{n^k})\\ &\to -\frac12(\ln(m)+\gamma+o(1))+\sum_{k=2}^{\infty}\frac{(-1)^{k}\zeta(k)}{k2^k}\\ &\to -\frac12(\ln(m)+\gamma)+C+o(1)\\ &\to -\frac12(\ln(m)+\gamma)+\frac12\gamma+ \frac12\ln(\pi) – \ln(2) +o(1)\\ &= -\frac12\ln(m)+ \frac12\ln(\pi/4) +o(1)\\ \end{array}$

since,
according to Wolfy,
$C =\sum_{k=2}^{\infty}\frac{(-1)^{k}\zeta(k)}{k2^k} = \frac12\gamma+ \frac12\ln(\pi) – \ln(2) \approx 0.167825$.

The sum to $2m+2$
would then be

$\begin{array}\\ -\frac12\ln(m)+ \frac12\ln(\pi/4)+ o(1)+\ln(2m+2) &=-\frac12\ln(m)+ \frac12\ln(\pi/4)+ o(1)+\ln(2)+\ln(m+1)\\ &=-\frac12 \ln(m)+ \frac12\ln(\pi/4)+ o(1)+\ln(2)+\ln(m)+\ln(1+1/m)\\ &=\frac12 \ln(m) + \frac12\ln(\pi)+o(1)\\ \end{array}$

Therefore the sum of
the even and odd sums
is
$\ln(\pi)-\ln(2)+o(1)$.
Therefore the average
of the first $m$ terms
goes to
$\frac12\ln(\pi/2) \approx 0.225791…$

So,
the Cesaro sum converges
so the limit exists
and the limit is
$\frac12\ln(\pi/2) \approx 0.225791…$

This post is to address partial progress(i.e not a full proof) of the conjecture, stated by @SimplyBeautifulArt in the comments secition using a weaker approach via the Borel Route, and Euler Summation.

$$\text{Euler Summation} \, \, (0.0):$$

One considers the Divigernt Series:

$$\sum_{}a_{n}$$

we replace our divigrent series with the corresponding power series:
$$\sum_{} a_{n}x^{n}$$
If such series is convergent for $|x| < 1$ and if it’s limit $x \rightarrow 1^{-}$ then one defines the Euler summation of the orginal series as:
$$\text{E}( \sum_{n} a_{n}) = \lim_{x \rightarrow 1^{-}} a_{n}x^{n}$$

$$\text{Proposition} (1.1)$$

$$\lim_{x\to-1^+}\sum_{n=2}^\infty x^n\ln(n)=\lim_{N\to\infty}\frac1N\sum_{k=2}^N\sum_{n=2}^k(-1)^n\ln(n)$$

Remark: We assume the RHS side converges

$$\text{Lemma}(0.1):$$

One can observed on our original Proposition the corresponding series on the RHS side can be rewritten as follows.

$$\lim_{x\to-1^+}\sum_{n=2}^\infty x^n\ln(n)= \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k}(-1)^{n} + \sum_{n} \ln(n)$$

Now focusing our observations on the RHS side of our recent result, another key consideration that can be made is by applying Borel Summability as follows in $(2.)$

$(2.)$

$$\lim_{x\to-1^+}\sum_{n=2}^\infty x^n\ln(n)= \lim_{n \rightarrow \infty} \frac{1}{N}(\lim_{t \rightarrow \infty} e^{-t} \sum_{k}\frac{t^{n}}{n!}(\sum_{n}(-1)^{n}) + \lim_{x \rightarrow 1^{-}}\sum_{n} \ln(x)^{n})$$$$\text{Remark}$$

The recent development seen in $(2.)$ can’t just be achived with Borel Summability alone the series $\sum\ln(n)$ was dealt with via Euler Summation as formally discussed in $(0.0)$. So considering the formalities of Euler Summation the series $\sum\ln(n)$ can be defined as follows:

$$\text{E}( \sum_{n} \ln(n)) = \lim_{x \rightarrow 1^{-}} \sum_{n} \ln(x)^{n})$$

@Simply can you continue from here 🙂

For $|z| \le 1, \Re(s)> 1$ and for $|z| < 1$ let the polylogarithm $$Li_s(z) = \sum_{n=1}^\infty n^{-s}z^n$$
For $\Re(s) > 1$ we have $Li_s(1) = \zeta(s), Li_s(-1) = -\eta(s)$.

Now for $|z| \le 1, z \ne 1$, summation by parts shows that $Li_s(z)$ is entire in $s$ and continuous in $z$ and hence $$-\eta(s) = \lim_{z \to -1, |z| \le 1} Li_s(z), \qquad -\eta'(s) = \lim_{z \to -1, |z| \le 1} \frac{\partial}{\partial s}Li_s(z)$$ $$\lim_{z \to -1, |z| \le 1} \sum_{n=1}^\infty z^n \log n = -\eta'(0) = \frac{\log(\pi/2)}{2}$$
(see there)