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The following power series apparently converges only for $-e \leq x <e$:

$$f(x)=\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$$

We can use it to define a real function $f(x)$, analytic in that interval.

- Gamma & Zeta Summation $\sum_{n=0}^{\infty}\frac{\Gamma(n+s)\zeta(n+s)}{(n+1)!}=0$
- Does the Abel sum 1 - 1 + 1 - 1 + … = 1/2 imply $\eta(0)=1/2$?
- Analytical continuation of moment generating function
- Gamma Infinite Summation $\sum_{n=0}^{\infty}\frac{\Gamma(n+s)}{n!}=0$
- Analytic continuation for $\zeta(s)$ using finite sums?
- Alternating series test for complex series

However, we can also use an integral to define this function:

$$f(x)=\int_0^\infty \frac{dt}{e^t-xt}=\sum_{k=1}^\infty \frac{(k-1)!}{k^k} x^{k-1}$$

In the interval of convergence of the series these two definitions are equivalent. However, for $x<-e$ the power series diverges, but the integral converges:

Can the integral serve as the analytic continuation of $f(x)$ for $x<-e$? How to justify this?

**Edit**

The integral works great for the complex plane as well. Here I used the integral representation to plot the real and imaginary parts of $f(z)$:

We have problems on the real line for $x>e$. For the power series – they converge on a disk $|z|<e$ (with the boundary possibly included):

- A difficult one-variable exponential integral
- Stuck in integration problem
- Closed-form of $\int_0^{\pi/4} \sin(\sin(x)) \, dx$
- Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \Bigl(\frac{\pi}{n\sin(\frac{\pi}{n})}\Bigr)^2$
- A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$
- Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$
- How to prove $\int_0^{2\pi} \ln(1+a^2+2a\cos x)\, dx=0$?
- Variable in Feynman Integration
- Closed-form of $\int_0^1\left(\frac{\arctan x}{x}\right)^n\,dx$
- How to prove Left Riemann Sum is underestimate and Right Riemann sum is overestimate?

- Algorithm to answer existential questions – Reduction
- If $r$ is a primitive root of odd prime $p$, prove that $\text{ind}_r (-1) = \frac{p-1}{2}$
- Compute integral closure of $F/(x^2-y^2z)$.
- Singular vector of random Gaussian matrix
- The system of genus characters determined by a binary quadratic form
- Primary ideals confusion with definition
- Finite union sigma field
- In Group theory proofs what is meant by “well defined”
- Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$
- The number of summands $\phi(n)$
- Relation between semidirect products, extensions, and split extensions.
- What will be the value of the following determinant without expanding it?
- How does trigonometric substitution work?
- Proof writing: how to write a clear induction proof?
- Intuitively, why is the Euler-Mascheroni constant near sqrt(1/3)?