7
votes
4answers
155 views

A closed form of $\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
1
vote
1answer
48 views

Limit of factorial how to continue

$$\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{(n+1)!-n!}}\right)=\lim_{n\to\infty}\left(\dfrac{(n+1)^{n+1}\cdot n!}{2^{n!\cdot n}}\right).$$ How to continue? the answer is $0$ ... thank you ...
4
votes
2answers
124 views

Limit of $a(k)$ in $ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $

For n = 1, 2, 3 ... (natural number) $ \sum_{k=1}^n \frac{a_k}{(n+1-k)!} = 1 $ $ a_1 = 1, \ a_2 = \frac{1}{2}, \ a_3 = \frac{7}{12} \cdots $ What is the limit of {$ a_k $} $ \lim_{k \to \infty} ...
1
vote
1answer
63 views

Proving $\int^\infty_0 x^n e^{-x} \, dx = n!$

I was motivated by this question on the various applications of integration by parts to prove the following integral: $$\int^\infty_0 x^n e^{-x} \, dx = n!$$ Here's what I have done, I feel I am ...
2
votes
2answers
83 views

How can I determine convergence of the series of $\frac {(2n)!}{2^{2n}(n!)^2}$

Does the series $$\sum_{n = 1} ^ {\infty} \frac {(2n)!}{2^{2n}(n!)^2}$$ converge? The ratio test doesn't work for the series.
0
votes
5answers
68 views

Prove that $\lim_{n \to \infty} \frac{n!}{n^n} = 0$

Prove that $\lim_{n \to \infty} \frac{n!}{n^n} = 0$ I've already considered using l'Hoptials rules but I cannot take the derivative of a factorial (as it is a discrete function). Thanks
1
vote
3answers
86 views

How to prove that $\lim_{n \to\infty} \to \frac{(2n-1)!!}{(2n)!!}=0$

So guys, how can I evaluate and prove that $\lim_{n \to\infty} \to \frac{(2n-1)!!}{(2n)!!}=0$. Any ideas are welcomed.
8
votes
3answers
122 views

Showing $n!<e(\frac{n}{2})^n$

I'd like to prove that $n!<e(\frac{n}{2})^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots +n}{n}=\frac{(n+1)n}{2n}=\frac{(n+1)}{2}$. Thus ...
1
vote
2answers
87 views

Can non-integer factorials be calculated without numerical integration?

I saw a strange way to write the factorial function somewhere and after some integration by parts, it all sure enough worked out. $$ n! = \int_0^\infty x^{n}e^{-x}dx $$ $$ ...
4
votes
3answers
119 views

Evaluating $\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$ [duplicate]

$$\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$$ Now, $\log(n!) = \Theta (n\log(n))$ so I think we could write, $$\lim_{n\to\infty}\frac{1}{2n}\left(\log\left(2n!\right) - ...
9
votes
1answer
166 views

Study of the convergence of a sequence with repeated radicals

Consider the sequence $$ a_n = \sqrt {1!\sqrt {2!\cdots\sqrt {n!} } }, \quad n\in\mathbb N. $$ Does this sequence converge? Clearly, $a_n$ is monotonically increasing. Therefore, there are two ...
3
votes
2answers
96 views

Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
3
votes
2answers
114 views

Limit of the sequence $\frac {a^n} {n!}$

I need to prove that $$\lim_{n \rightarrow \infty} \frac {a^n} {n!}=0$$ I have no condition over $a$, just that is a real number. I thought of using L'Hôpital, but it's way too complicated for ...
0
votes
3answers
99 views

Estimate the factorial $n!$ starting with the integral of $1/x$

This is a 3-part problem concerning an estimate for the factorial $n!$ a. By considering the graph of $y=\frac{1}{x}$ explain why $$\frac{1}{k+1} < \int\limits_{k}^{k+1} \frac{\mathrm ...
0
votes
1answer
45 views

Review of an answer for finding a limit of a sequence

$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } {{n!} \over {(n + 1)(n + 2)...(2n)}} = {{n!} \over {{{(2n)!} \over {n!}}}} = \cr & {{n!n!} \over {2n!}} = {{n!} \over 2} = + \infty ...
1
vote
1answer
31 views

Is there an actual expansion of the Gamma function's integral?

$$\int_0^{\infty} x^{t-1} e^{-x} \, \mathrm{d}x = (t-1)! = \Gamma (t)$$ Is the expression $(t-1)!$ the actual result of integrating the gamma integral? Meaning, if you were to compute the integral ...
0
votes
3answers
56 views

Is $\sum_{n=1}^{\infty}\frac{2^nn!}{(n+1)!}$ absolutely convergent?

I'm very uncomfortable with factorials just because I haven't done many of them. But my basic understanding is if I start with (for example) $(n+1)!$ then this is equivalent to $(n+1)*(n)$ and if it ...
0
votes
1answer
34 views

Series — Coefficient Cn and Radius of Convergence

. I'm lost, and my textbook is failing me
0
votes
2answers
38 views

For $j \in \{0,…,n-1\}$ is $(n-j)!(j+1)! \leq n!$ true?

For $j \in \{0,...,n-1\}$ is $(n-j)!(j+1)! \leq n!$ true? I mean $\dfrac{n!}{(n-j)!(j+1)!}$ doesn't have to be an integer. I need this inequality in another exercise, so Is it provable?
1
vote
1answer
71 views

How to derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n+ 1)$?

How can you derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n + 1)$? I have tried checking Wolfram Alpha for a step-by-step solution, but none is given. Moreover, of what is the second function, ...
0
votes
1answer
65 views

Finite calculus: Apply difference operator to generalized falling factorial $(ax+b)^{\underline m}$

The $m$th falling factorial power of $x$ is defined as $x^{\underline m}:=x(x-1)...(x-m+1),$ and the difference operator as $\Delta f(x) := f(x+1)-f(x).$ One fundamental statement in finite ...
3
votes
1answer
1k views

Using the Squeeze Theorem in Sequences

My textbook has an example that says "Show that the sequence {${c_n}$} $= (-1)^n \frac{1}{n!} $ " converges, and find its limit. It tells me that I must "find two convergent sequences that can be ...
4
votes
2answers
541 views

Proof that the gamma function is an extension of the factorial function

I've already proved that $$\Gamma (n)= (n-1)!$$ but I don´t really know what else to do to verify that $\Gamma$ is an extension of the factorial function for real numbers (positive) Thank you! And ...
1
vote
0answers
58 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
3
votes
1answer
703 views

Limit of the sequence $\{n^n/n!\}$

I am trying to check whether or not the sequence $$a_{n} =\left\{\frac{n^n}{n!}\right\}_{n=1}^{\infty}$$ is bounded, convergent and ultimately monotonic (there exists an $N$ such that for all $n\geq ...
2
votes
1answer
196 views

How do you solve an inequality with the factorial of a variable?

How do you solve an inequality with the factorial of a variable? Example: Determine the interval of $n \in \Bbb N$ for which the following inequality holds: $$n! \leq 157788 \cdot 10^{10} $$ Can ...
6
votes
3answers
148 views

Proving a complex sum equals factorial

I have just stumbled across the equality that: $$ \sum_{j=0}^{n}(-1) ^ {n + j} j ^ {n} \binom{n}{j} = n! $$ How would I go about proving this equality? Also, what is the left hand side equal to if ...
1
vote
0answers
107 views

Double summation including power and factorial [duplicate]

I am finding some trouble in computing the following sum: $$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac {y^m}{m!}$$ Could you please provide a result? Thanks in advance
1
vote
3answers
900 views

limits of sequences exponential and factorial

Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences: (a) $a_n=e^{5\cos((\pi/6)^n)}$ (b) $a_n=\frac{n!}{n^n}$ For part (a) do I just take the limit of the exponent part and ...
1
vote
1answer
152 views

Gamma function question

Gamma function is also known as generalized factorial function . why does the term "generalized" have been used ? Again, why is the Gamma function called Euler's second integral?
5
votes
1answer
237 views

Sum involving the hypergeometric function, power and factorial functions

I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions. $$ \sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
4
votes
3answers
865 views

Calculating limit involving factorials.

I want to show that $\lim\limits_{k\to\infty} \frac{\pi^kk!}{(2k+1)!} = 0$. I've been trying to use the squeeze theorem, but am having a hard time finding some expression $P$ involving $k$ that is ...
3
votes
4answers
12k views

Derivative of a factorial

What is ${\partial\over \partial x_i}(x_i !)$ where $x_i$ is a discrete variable? Do e consider $(x_i!)=(x_i)(x_i-1)...1$ and do product rule on each term, or something else? THanks.
5
votes
1answer
239 views

A limit involves series and factorials

Evaluate : $$\lim_{n\to \infty }\frac{n!}{{{n}^{n}}}\left( \sum\limits_{k=0}^{n}{\frac{{{n}^{k}}}{k!}-\sum\limits_{k=n+1}^{\infty }{\frac{{{n}^{k}}}{k!}}} \right)$$
4
votes
6answers
2k views

How can I calculate the limit of exponential divided by factorial? [duplicate]

I suspect this limit is 0, but how can I prove it? $$\lim_{n \to +\infty} \frac{2^{n}}{n!}$$
1
vote
4answers
169 views

Find $\sum_{n=1}^{\infty}\frac{1}{n!}$

Find $$\sum_{n=1}^{\infty}\frac{1}{n!}$$ All of the advice I've seen to compute this sum says to use the ratio test, but this is in a chapter BEFORE the ratio test, so the book wants me to solve ...
0
votes
2answers
363 views

Derivative of Gauss' Pi Function

Famous mathematician and scientist Carl Gauss developed the Pi function $$\Pi(x)=\int_{t=0}^\infty t^x e^{-t}\,dt$$ which has the property that ...
1
vote
3answers
3k views

Find the limit of exponent/factorial sequence [duplicate]

Possible Duplicate: Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$. Finding $\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$ I don't know how to even stoke it... ...
3
votes
3answers
249 views

How to approximate $\sum_{k=1}^n k!$ using Stirling's formula?

How to find summation of the first $n$ factorials, $$1! + 2! + \cdots + n!$$ I know there's no direct formula, but how can it be estimated using Stirling's formula? Another question : Why can't ...
4
votes
2answers
152 views

Upper bound for the series $\sum_{n\geq 1}\frac{1}{(n+1)^{a+1}}\sum_{k=0}^n b^k\left(\frac{(n-k)!}{n!}\right)^a$

I want to show that the series $$\sum_{n\geq 1}\frac{1}{(n+1)^{a+1}}\sum_{k=0}^n b^k\left(\frac{(n-k)!}{n!}\right)^a$$ converges for $a,b>0$. I have tried this so much that the smallest hint will ...
2
votes
1answer
42 views

Limit of $\frac{a^{n+1}(n+1)!^b}{\sum_{k=0}^n a^kk!^b}$ when $n\rightarrow\infty$

I want to prove that $$\lim_{n\rightarrow\infty}\frac{a^{n+1}(n+1)!^b}{\sum_{k=0}^n a^kk!^b}<\infty$$ for $a,b>0$. This is the last step of a bigger problem. I believe it would suffice to use ...
1
vote
1answer
105 views

Determining the type of convergence of a series

The specific homework problem is $\displaystyle\sum\limits_{n=0}^\infty \frac{2\cdot 4\cdot 6\cdots2n}{n!}$. It is problem #29 from the section 11.6 exercises (pg. 737) from Single Variable Calculus: ...
13
votes
6answers
757 views

A question on the Stirling approximation, and $\log(n!)$

In the analysis of an algorithm this statement has come up:$$\sum_{k = 1}^n\log(k) \in \Theta(n\log(n))$$ and I am having trouble justifying it. I wrote $$\sum_{k = 1}^n\log(k) = \log(n!), \ \ ...
9
votes
5answers
771 views

How to prove that $\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2)… 2n} = \frac{4}{e}$

I'd like a hint to show that: $$\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2) \cdots 2n} = \frac{4}{e} .$$ Thanks.