3
votes
2answers
85 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
55 views

Prove that $ \left\{\frac{\binom{n+k}{k} }{(n+k)^k}\right\}_{n=1}^{\infty} $ converges to $\frac{1}{k!}$.

Prove that $$ \left\{\frac{\binom{n+k}{k}}{(n+k)^k}\right\}_{n=1}^{\infty} $$ converges to $\frac{1}{k!}$, where $$\binom{n+k}{k} =\frac{(n+k)!}{n!k!}. $$ I'm not even sure how to approach ...
3
votes
0answers
55 views

Question about factorial function [duplicate]

Show that $$n!=1+\left(1−{1 \over 1!}\right)n+\left(1−{1 \over 1!}+ {1 \over 2!}\right)n(n−1)+\cdots$$ I can't figure out how this can be solved . I tried to use binomial theorem but I couldn't prove ...
3
votes
2answers
64 views

Simple factorial function question

Show that $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+....$ I can't figure out how this can be solved . I tried to use binomial theorem but I couldn't prove it . Any help will be greatly appreciated.
2
votes
2answers
81 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.
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 ...
8
votes
1answer
151 views

Study of the convergence of a sequence with repeated radicals

Let the sequence $$ a_n = \sqrt {1!\sqrt {2!\cdots\sqrt {n!} } } $$ Does this sequence converge? I can tell intuitively that $a_n$ is monotonically increasing. Therefore, there are two ...
3
votes
0answers
57 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
1
vote
1answer
108 views

Stirling approximation / Gamma function

Is it possible to obtain the Stirling approximation for the factorial by using the gamma function ? $$\Gamma(z) = \lim_{n \to +\infty} \frac{n! n^{z}}{z(z+1) \dots (z+n)}$$ Any hint ?
3
votes
2answers
111 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 ...
5
votes
1answer
50 views

limit of sequence with factorial

How do you show that: $\lim\limits_{n\to \infty} \frac{\left(\frac{n}{2}\right)^{\frac{n}{2}}}{n!}=0$ using the squeeze theorem (I'd like to avoid using Stirling's formula, too). I tried rearranging ...
0
votes
1answer
39 views

Summation of a curious series-repeated division by primes

I am interested in knowing if there is some closed form/formula for the following series: ...
2
votes
1answer
292 views

Infinite Sum of factorial denominator and exponential numerator

I've been trying to find the sum of the following infinite series: $$ \sum\limits_{n=1}^\infty \frac{x^n}{n!2^n} $$ I've rewritten it as $$\sum\limits_{n=1}^\infty \frac{y^n}{n!}, y=\frac{x}{2}$$ ...
2
votes
1answer
101 views

Why doesn't $0! = 1$ in the context of this general term?

Is my instructor wrong to say that $\left\{0,\frac{1!}{4},\frac{2!}{9},\frac{3!}{16},\dots\right\} = \left\{\frac{(n-1)!}{n^2}\right\}$? My understanding is that at $n=1$, $\frac{(n-1)!}{n^2}$ should ...
2
votes
1answer
82 views

Convergence of $\sum\limits_{n=1}^\infty \frac{n!}{n^n} \times (5x)^n$

I have to check for which $x$ the series converges/diverges. $\sum\limits_{n=1}^\infty\frac{n!}{n^n} \times (5x)^n$ I know that for $|x| < \frac{1}{5}e$ it converges and for $|x| > ...
2
votes
3answers
84 views

Limit of sequence. with Factorial

Can't find the limit of this sequence : $$\frac{3^n(2n)!}{n!(2n)^n}$$ tried to solve this using the ratio test buy failed... need little help
0
votes
1answer
113 views

Limit of a fraction of double factorials

How can we show that $\begin{align*} \lim_{n\rightarrow\infty} U_n = 0 \end{align*}$ where $\begin{align*} U_n = \frac{(n-1)!!}{n!!}=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4}\cdots \end{align*}$ ...
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
3
votes
5answers
192 views

Need help understanding the factorial formula $n!=n(n-1)(n-2)\cdots(3)(2)(1)$

The caption says the following: If $n$ is an integer such that $n \ge 0$ then $n$ factorial is defined as, $$n!=n(n-1)(n-2)\cdots(3)(2)(1)$$ if $n \ge 1$ by definition. I'm really just confused by ...
0
votes
1answer
63 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
59 views

Simplifying this logarithm series

$$\sum_{i\; =\; 2}^{99}{\frac{1}{\log _{i}\left( 99! \right)}}$$ How would you evaluate (or at least simplify) this logarithm series?
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 ...
1
vote
0answers
52 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: ...
1
vote
2answers
816 views

the nth root of n!?

I am playing around with the root/ratio test to practice with series. I just showed that $\sum \frac{1}{n!}$ converges by using the ratio test. I decided to see how things would go with the root test ...
1
vote
0answers
63 views

Limit of the sequence $\frac{(n!)^{1/n}}{n}$ [duplicate]

Which is the limit of the fllowing sequence $$\frac{(n!)^{1/n}}{n}$$
6
votes
4answers
483 views

Find the sum of series $\sum_{n=0}^\infty\frac{(4n)!}{(4n+4)!}$

I wanted to know how can I start to find the sum of the series: $$\sum_{n=0}^\infty\frac{(4n)!}{(4n+4)!}=\frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}\cdots$$ I am having no clue. Thanks.
1
vote
1answer
51 views

At what $n$ does $\sum_{k=0}^\infty{(n\ k)!\over (k!)^3}$ diverge?

Consider $$\sum_{k=0}^\infty{(n\ k)!\over (k!)^3}$$ where $0<n$. For what $x$ where $0<x\leq n$ does the sum diverge?
12
votes
1answer
319 views

Yet another $\sum = \pi$. Need to prove.

How could one prove that $$\sum_{k=0}^\infty \frac{2^{1-k} (3-25 k)(2 k)!\,k!}{(3 k)!} = -\pi$$ I've seen similar series, but none like this one... It seems irreducible in current form, and I have ...
1
vote
1answer
299 views

Summation with factorial terms (involving Laguerre polynomials)

As part of an exercise including gamma functions and Laguerre polynomials, I need to show that for a Laguerre polynomial $L_n(x)$, $$\int\limits_0^\infty L_n(x)x^ke^{-x}dx = 0 \textrm{ with } n \in ...
1
vote
3answers
810 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
3answers
550 views

Testing for convergence in Infinite series with factorial in numerator

I have the following infinite series that I need to test for convergence/divergence: $$\sum_{n=1}^{\infty} \frac{n!}{1 \times 3 \times 5 \times \cdots \times (2n-1)}$$ I can see that the denominator ...
5
votes
1answer
234 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
2answers
148 views

Series involving factorials

How would one go about proving $$\int_{0}^1\frac{e^x-1}{x/2}\ dx=\sum_{n=0}^\infty\frac{1}{\binom{n+2}{2}}\frac{1}{n!}(0!+1!+2!+3!+...+n!)$$
3
votes
3answers
107 views

Evaluate $\sum_{k=1}^{n}(k^2 \cdot (k+1)!)$

We have to evaluate the following: $$1^2 \cdot 2! + 2^2 \cdot 3! + \cdots + n^2 \cdot (n+1)! =\sum_{k=1}^{n} k^2 \cdot (k+1)!$$ Any hints ?
1
vote
2answers
2k views

Summation of Series with Factorials

It is given that: $$v_n = n(n+1)(n+2)\;...\;(n+m)$$ $$and$$ $$u_n = (n+1)(n+2)\;...\;(n+m)$$ $i.$ Verify that: $$v_{n+1} - v_n = (m+1)(n+1)(n+2)\;...\;(n+m)$$ I started off by inspecting ...
19
votes
4answers
578 views

Limit of series involving ratio of two factorials

$$ \sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3} $$ The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
5
votes
1answer
236 views

A limit involves series and factorials

Evaluate : $$\lim_{n\to \infty }\frac{n!}{{{n}^{n}}}\left( \sum\limits_{k=0}^{n}{\frac{{{n}^{k}}}{k!}-\sum\limits_{k=n+1}^{\infty }{\frac{{{n}^{k}}}{k!}}} \right)$$
5
votes
2answers
398 views

How to simplify this equality (factorials)?

This was in one of the examples of the textbook, but I couldn't figure out how they solved it. They say they multiply the left hand side by $\frac{n!}{n!}$ to get the right hand side: $$ \frac{2^n ...
3
votes
6answers
252 views

Show that the sequence $\left(\frac{2^n}{n!}\right)$ has a limit.

Show that the sequence $\left(\frac{2^n}{n!}\right)$ has a limit. I initially inferred that the question required me to use the definition of the limit of a sequence because a sequence is convergent ...
25
votes
2answers
4k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
3
votes
3answers
480 views

Infinite Series using Falling Factorials

I recently started reading Concrete Mathematics by Graham, Knuth and Patashnik and met falling/rising factorials for the first time; it seemed like a very convenient method for evaluating particular ...
5
votes
7answers
201 views

Some trouble with a proof on $n!/(\sqrt{n})^n \geq 1$

Originally the problem is to prove that $n! \geq n^{n/2}$. I reduced this to: $n! \geq (\sqrt{n})^n$ so that: Prove that $\frac{n!}{(\sqrt{n})^n} \geq 1$. Each term in $n!$ is divided by the ...
1
vote
3answers
120 views

Closed form for $T(1) = K, T(x) = xT(x-1) + x$?

I'm looking for a closed form for the following recurrence: $T(1) = K$ $T(x) = xT(x-1) + x$ I know it is factorial-like but I am unable to get an exact answer.
6
votes
3answers
127 views

closed-form expressions for product of 3n+k where k = 1 or 2

There are some easy products that can be written in closed form in terms of factorials: $ 2 \times 4 \times 6 \times ... 2n = n! \times 2^n$ $ 1 \times 3 \times 5 \times ... (2n-1) = {{(2n)!} ...
4
votes
3answers
2k views

Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$

I need to check if $$\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$$ converges or not. Additionally, I wanted to show that the sequence is monotonically increasing in n and so limit exists. Any help is ...
1
vote
3answers
298 views

Summation of a factorial (total number terms in a polynomial)

By induction I can prove : $$\sum^{M}_{t=0}\frac{(t+D-1)!}{t!(D-1)!} = \frac{(D+M)!}{D!M!} $$ However, I couldn't derive the right hand side directly. It would be of great help if anyone can solve ...
4
votes
2answers
150 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 ...
3
votes
3answers
612 views

Notation for factorial-type pattern with a skip/step of two instead of one?

I came across a peculiar pattern when solving a recurrence relation today: Some sequence $a_n$ looks as such: $a_0 = 1$ $a_2 = \frac{1}{2 \cdot 1}$ $a_4 = \frac{1}{4 \cdot 2 \cdot 1}$ $a_6 = ...
7
votes
5answers
572 views

Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]

Possible Duplicate: Summation of a factorial This equation is given: $$ 1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1 $$ I've solved it using mathematical induction but ...