Intereting Posts

Deriving Taylor series without applying Taylor's theorem.
Total Variation and indefinite integrals
How to show that if all fourier coefficient of a function is zero, then the function is zero function?
solution of ordinary differential equation $x''(t)+e^{t^2} x(t)=0, for t\in$
Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued
The residue at $\infty$
Find the coordinates in an isosceles triangle
let $H\subset G$ with $|G:H|=n$ then $\exists~K\leq H$ with $K\unlhd G$ such that $|G:K|\leq n!$ (Dummit Fooote 4.2.8)
Cross product in $\mathbb R^n$
Prove that convex function on $$ is absolutely continuous
Any quotient group of $(\mathbb Q,+)$ is torsion
Bijection from $\mathbb{R}^n$ to $\mathbb{R}$ that preserves lexicographic order?
Questions about finite sequences of natural numbers with distinct partial sums
In a group of exponent $2^n$, $=1$?
The square root of a prime number is irrational

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,

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

- does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?
- Cluster Point in Fréchet-Urysohn Space is Limit Point?
- Does the series $\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?
- Confused by the Bolzano Weierstrass Theorem
- What is the difference between the limit of a sequence and a limit point of a set?
- Subsequence convergence in $L^p$

What are my alternatives?

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

- Harmonic number divided by n
- If $\lim_n f_n(x_n)=f(x)$ for every $x_n \to x$ then $f_n \to f$ uniformly on $$?
- Approximation of conditional expectation
- Am I right in my conclusions about these series?
- Almost sure convergence of random variables
- Proof that a sequence converges to a finite limit iff lim inf equals lim sup
- Is this a valid proof for ${x^{x^{x^{x^{x^{\dots}}}}}} = y$?
- If $\sum_{n=1}^\infty \frac{1}{a_n}$ converges, must $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ converge?
- Taylor series expansion and the radius of convergence
- Ratio test with limsup vs lim

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

- Race Problem counting
- Which $f \in L^\infty$ are the Fourier transform of a bounded complex measure?
- How to prove the distributive property of cross product
- Studying for the Putnam Exam
- Interchanging pointwise limit and derivative of a sequence of C1 functions
- Continuous increasing bounded function, derivative
- GCD computations in $\mathbb{Z}$
- Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
- Finding limits using definite integrals $\lim_{n\to\infty}\sum^n_{k=1}\frac{k^{4}}{n^{5}}$
- When is a finite sum of powers of non-integer a rational number?
- What is category theory?
- Independent increments in squared brownian motion
- Expected Values of Operators in Quantum Mechanics
- Weak limit and strong limit
- Proving that $C$ is a subset of $f^{-1}$