Intereting Posts

Explain inequality of integrals by taylor expansion
Energy functional in Poisson's equation: what physical interpretation?
Complex Galois Representations are Finite
Is memorization a good skill to learn or master mathematics?
What is wrong with this effort to generalize Bertrand's Postulate using the Inclusion-Exclusion Principle
Embeddability of the cone of Klein bottle to $\mathbb R^4$
How to find the vertices angle after rotation
Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
Physical or geometric meaning of complex derivative
Is “A New Kind of Science” a new kind of science?
Changing limits in absolutely convergent series
Why is there this strange contradiction between the language of logic and that of set theory?
Finding the elements of a finite field?
Equivalence of Brouwers fixed point theorem and Sperner's lemma
Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$

I’m working on a problem sheet and it ask to discuss the convergence of

$$\sum \frac{n!}{{n}^{n}}$$

By D’Lembert’s ratio test,

$$\lim_{n->\infty}\frac{{a}_{n+1}}{{a}_{n}} = 1$$

and so, is inconclusive.

Using Cauchy’s root test,

$$\lim_{n->\infty}({\frac{n!}{{n}^{n}}})^\frac{1}{n}=1$$

- What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
- Taylor series expansion and the radius of convergence
- Alternative formulation of the supremum norm?
- Convergence of a sequence and some of its subsequences
- How to prove convergence of polynomials in $e$ (Euler's number)
- Convergence of Roots for an analytic function

What are my alternatives?

Should I take the integral of the term of the series above? Would integrating factorial works?

- convergence of the iterated cosine
- Weak convergence in probability implies uniform convergence in distribution functions
- Difference of differentiation under integral sign between Lebesgue and Riemann
- Divergent or not series?
- Elementary proof, convergence of a linear combination of convergent series
- What is the difference between the limit of a sequence and a limit point of a set?
- Interchanging pointwise limit and derivative of a sequence of C1 functions
- Inequality regarding norms and weak-star convergence
- Rearrangement of double infinite sums
- Show $\lim_{n \to \infty} \sum_{i=1}^n Y_i/\sum_{i=1}^n Y_i^2 = 1$ for Bernoulli distributed random variables $Y_i$

Actually the ratio test turns out to be conclusive :

$$\begin{align}\lim_{n\to\infty}\frac{{a}_{n+1}}{{a}_{n}} &=\lim_{n\to\infty}\dfrac{(n+1)!}{(n+1)^{n+1}} \cdot \dfrac{n^n}{n!} \\~\\&=\lim_{n\to\infty}\dfrac{n+1}{(n+1)^{n+1}} \cdot \dfrac{n^n}{1} \\~\\&=\lim_{n\to\infty} \left(\dfrac{n}{n+1}\right)^n\\~\\&=\lim_{n\to\infty} \left(\dfrac{\color{blue}{n+1}-1}{n+1}\right)^n\\~\\&=\lim_{n\to\infty} \left(\color{blue}{1}+\dfrac{-1}{n+1}\right)^n\\~\\&=e^{-1}~~\color{Red}{\star} \\~\\&\lt 1\end{align}$$

$\color{red}{\star}$ : please see $e^x$ limit definition

$$\frac{n!}{n^n}=\frac{1}{n}\frac{2}{n}\cdots\frac{n-1}{n}\frac{n}{n}<\frac2{n^2}$$

**Hint**: $\dfrac{n!}{n^n} < \dfrac{2}{n^2}$.

- Best book on axiomatic set theory.
- Least area of maximal triangle inside convex $n$-gon.
- Necessary and sufficient condition for $R/IJ \cong R/I \times R/J$
- Interpretations of the first cohomology group
- Local vs global truncation error
- evaluate the last digit of $7^{7^{7^{7^{7}}}}$
- Prove that $\gcd(a^n – 1, a^m – 1) = a^{\gcd(n, m)} – 1$
- Find a closed form of the series $\sum_{n=0}^{\infty} n^2x^n$
- Converting a Gomoku winning strategy from a small board to a winning strategy on a larger board
- Difference between the formula of Roger Cotes and Euler
- How to integrate $\int \frac{e^x dx}{1\,+\,e^{2x}}$
- An Explanation of the Kalman Filter
- Show that an infinite number of triangles can be inscribed in either of the parabolas $y^2=4ax$ and $x^2=4by$ whose sides touch the other parabola.
- Conditions on a $1$-form in $\mathbb{R}^3$ for there to exist a function such that the form is closed.
- Problem with the ring $R=\begin{bmatrix}\Bbb Z & 0\\ \Bbb Q &\Bbb Q\end{bmatrix}$ and its ideal $D=\begin{bmatrix}0&0\\ \Bbb Q & \Bbb Q\end{bmatrix}$