$c_n = 1 + \frac{1}{1!} + \frac{1}{2!} +...+\frac{1}{n!}$, so $e - c_n \le \frac{1}{n! * n}$  I absolutely have no idea how to solve it, could anyone tell me the approach?
107339
Prove that:  $$\displaystyle \sum_{n=m+1}^\infty \dfrac{1}{n!} \le \dfrac{1}{m\cdot m!}$$  I have tried induction on $m$ but it does not work very well. Any suggestion?
49802
I used $$(n!)^{\frac{1}{n}}=e^{\frac{1}{n}\ln(n!)}=e^{\lim\limits_{n\to\infty}\frac{1}{n}\ln(n!)}$$ Then using Stirling's approximation and L'Hospital's rule on $$\lim\limits_{n\to\infty}\frac{\ln(n!)}{n}$$ I get $$\lim\limits_{n\to\infty}\frac{\ln(n!)}{n}=\lim\limits_{n\to\infty}(\ln(n)+\frac{n+\frac{1}{2}}{n}-1)=\infty$$ Now, $$e^{\lim\limits_{n\to\infty}\frac{1}{n}\ln(n!)}=e^{\infty}=\infty$$ Thus $$\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]n}=\frac{1}{\infty}=0$$  Is this correct approach and what other methods could be used?
15251
How can the following equation be proven?     $$ \forall n &gt; 2 : \sum_{p \le n}{\frac1{p}} = C + \ln\ln n + O\left(\frac1{\ln n}\right), $$  where $p$ is a prime number.   It's not homework; I just don't understand from where should I start.
74594
Using permutation or otherwise, prove that $\displaystyle \frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.  I have no idea how to prove this..!!I am not able to even start this Can u give some hints or the solution.!cheers.!!
30374
I have to calculate sum of series $\sum \frac{n^2}{n!}$. I know that $\sum \frac{1}{n!}=e$ but I dont know how can I use that fact here..
85477
For all positive integers $n,r,k$ that $n=rk$, how to prove the inequality $$(r-1)!^k\le \frac{c(n,k)}{S(n,k)}\le (n-k)!$$ $c(n,k)$ is unsigned Stirling numbers of the first kind and $S(n,k)$ is Stirling numbers of the second kind.  Any help would be appreciated!
76207
Can someone please explain to me how is $$\dfrac1{(1-x)^5}=\sum_{n\geq0}{n+4\choose4}x^n$$  Thanks!
40315
$$\frac{1}{1 \cdot 2} + \frac{1}{2\cdot 3} + \cdots + \frac{1}{n(n+1)} = \frac{n}{n+1}$$  What I have so far in the induction is:  $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{k(k+1)} + \frac{1}{(k+1)(k+2)} = \frac{k}{k+1} + \frac{1}{(k+1)(k+2)}$$  But I'm not sure where to go from there . . .
114797
Question is to Evaluate :  $$\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots$$  what all i could do is :  $$\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots=\sum_{n=1}^{\infty} \frac{1}{(n+2)n!}=\sum_{n=1}^{\infty} \frac{n+1}{(n+2)!}=\sum_{n=1}^{\infty} \frac{n}{(n+2)!}+\sum_{n=1}^{\infty} \frac{1}{(n+2)!}$$  I have $$\sum_{n=1}^{\infty} \frac{1}{(n+2)!}=\sum_{n=0}^{\infty} \frac{1}{n!}-1-\frac{1}{2}=e-\frac{3}{2}$$  Now, I am not able to see what $$\sum_{n=1}^{\infty} \frac{n}{(n+2)!}$$ would be.  I would be thankful if some one can help me to clear this.  Thank you.
62356
Prove that $ \frac{n(n-1)(n-k+1)}{k(k-1)\cdots1} = \frac{n!}{(n-1)!k!}$.   Proof:  This is what I have tried:  $\frac{n!}{k!(n-k)!} \times (n-k)!  = \frac{n(n-1)\cdots(n-k+1)(n-k)!}{k(k-1)\cdots1(n-k)!}$  $\frac{n!}{k!} = \frac{n(n-1)(n-2)\cdots(n-k+1)}{k(k-1)\cdots1}$  This is where my attempt ends.   Are my steps correct? If not how can I improve them?  New Questions:  How is it that n-k magically appears on the right side? I get that n-k and 3 times times 2 times 1 cancels out in the numerator and the denominator which leaves   $\frac{n(n-1)(n-2)...(n-k+1)}{k(k-1)....1}$ which is equivalent to ${n \choose k}$ or \frac{n!}{(n-k)!k!}....is that correct?
17841
I tried and got this   $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an integer. $$\lim_{n\to\infty}n\sin(2\pi en!)=\lim_{n\to\infty}n\sin\left(2\pi n!\sum_{k=0}^n\frac{1}{k!}\right)=\lim_{n\to\infty}n\sin(2\pi m)=\lim_{n\to\infty}n\cdot0=0$$  Is it correct?
17910
Show that $\displaystyle{n!=1+\left(1-\frac1{1!}\right)n+\left(1-\frac1{1!}+\frac1{2!}\right)n(n-1)+\cdots}$. I can't figure out how this can be solved. I tried to use the binomial theorem but I couldn't prove it. Any help will be greatly appreciated.
31850
Show that $$n! \leqslant \left(\frac{n+1}{2}\right)^n \quad \hbox{for all } n \in \mathbb{N}$$  I know that it can be done by induction but I always find line where I do not know what to do next.
86636
If $a^2+b^2+c^2=1$, prove that $\frac{-1}{2}\le\ ab+bc+ca\le 1$.  I was able to prove that $ ab+bc+ca\le 1$. But I am unable to gain an equation to prove that $ \frac{-1}{2}\le\ ab+bc+ca$ .  Thanks in advance !
54720
Mathematica tells me that  $$\sum _{k=0}^n { n \choose k} \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}.$$  Although I have not been able to come up with a proof.  Proofs, hints, or references are all welcome.
64569
I have to show that if $|z| &lt; 1$, $z \in \mathbb{C}$, $$(1+z)\prod_{n=1}^{+\infty}(1+z^{2^{n}})= \frac{1}{1-z}$$  I want to understand how to do this kind of exercises, any hint ?
60943
If $a,b,c &gt; 0$ are such that $abc=1$, then   $$ \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c. $$   I would be pleased if you give me a hint. Thanks in advance.