# Linked Questions

51 questions linked to/from Stirling's formula: proof?
2answers
552 views

### Is there a “simple” way of proving Stirlings formula? [duplicate]

Is there any way to derive Stirlings formula that only requires some undergraduate knowledge of calculus, real analysis and perhaps some identitets involving the gamma function, maybe Wallis product, ...
2answers
62 views

### Estimate how much large is $\lfloor \log{n!\rfloor}$ [duplicate]

Is there a simple method to find or estimate how large $$\lfloor \lg{n!\rfloor}$$ is ? I'd like to find (or estimate) how much digits $2017!^{2017}$ has, or how much is big that number . I tried ...
0answers
40 views

### Limit of Stirling's approximation as n goes to infinity. [duplicate]

I would like to see some detailed solution for $$\frac{n!}{\sqrt{2\pi n} \left(\frac{n}{e}\right)^n}$$ as $n\to\infty$. I know that the answer is 1 but i am not sure why? Here is what is tried: I ...
43answers
96k views

### Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
28answers
94k views

### Evaluating the integral $\int_0^\infty \frac{\sin x} x \ dx = \frac \pi 2$?

A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral: $$\int_0^\infty \frac{\sin x} x \, dx = \frac \pi 2$$ Well, can anyone prove ...
4answers
6k views

### Tough integrals that can be easily beaten by using simple techniques

This question is just idle curiosity. Today I find that an integral problem can be easily evaluated by using simple techniques like my answer to evaluate \begin{equation} \int_0^{\pi/2}\frac{\cos{x}}{...
4answers
9k views

### Summation of logs

Are there any useful identities for quickly calculating the sum of consecutive logs? For example $\sum_{k=1}^{N} log(k)$ or something to this effect. I should add that I am writing code to do this (as ...
4answers
453 views

### Prove that $1.49<\sum_{k=1}^{99}\frac{1}{k^2}<1.99$

It can be proven by induction that $$\sum_{k=1}^{n}\frac{1}{k^2}\leq2-\frac{1}{n}$$ From here, we can easily acquire the upper bound of the sum $$\sum_{k=1}^{99}\frac{1}{k^2}$$ letting $n=100$. ...
3answers
275 views

### Evaluate the limit $\lim_{n\to\infty}\frac{n}{\ln n}\left(\frac{\sqrt[n]{n!}}{n}-\frac{1}{e}\right)$

Evaluate $$\lim_{n\to\infty}\frac{n}{\ln n}\left(\frac{\sqrt[n]{n!}}{n}-\frac{1}{e}\right).$$ This sequence looks extremely horrible and it makes me crazy. How can we evaluate this?
5answers
174 views

3answers
289 views

### Stirling approximation note

During my study to Stirling approximation I find this formula $n! \approx \sqrt{2\pi n} n^{n}e^{-n}$ but we know that $0! =1$ And in this formula if we replace every $n$ with $0$ we will ...
4answers
952 views

Let $c>0$ be a real number. I would like to study the convergence of $a_n := c^n n!/n^n$, when $n \to \infty$, in function of $c$. I know (from this question) that $n!>(n/e)^n$, so that $c^n n!/... 5answers 128 views ### Proof that$n!\leq {(\frac{n+1}{2})}^{n}$[duplicate] I don't know what to do with this. Nothing works. I hope somebody can help me to find a decision $$n!\leq {\left(\frac{n+1}{2}\right)}^{n}$$ 6answers 439 views ### If$n$is a positive integer, Prove that$\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac{2329}{3600}.$If$n$is a positive integer, Prove that $$\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac{2329}{3600}.$$ please don't refer to the famous$1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6\$. I am ...

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