In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n:
For example,
The value of 0! is 1, according to the convention for an empty product.The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there are n!.
The factorial function can also be extended to non-integer arguments while retaining its most important properties by defining x! = Γ(x + 1), where Γ is the gamma function; this is undefined when x is a negative integer.
I understand that the "spiral" converges to 1+i-1/2-i/3!+1/4!+i/5!-1/6!-i/7!... .
It splits into two: one for Re, 1-1/2+1/4!-1/6!..., and the other for Im, 1-1/3!+1/5!-1/7!... .
Any hints on how to compute them?
Without using computer programs, can we find the last non-zero digit of $$(\dots((2018\underset{! \text{ occurs }1009\text{ times}}{\underbrace{!)!)!\dots)!}}$$?
What I know is that the last non-zero digit of ##2018!## is ##4##, but I do not know what to do with that ##4##.
Is it useful that...
This may have already been found by many people but I discovered the pattern on my own out of curiosity with some coding.
There are only 4 natural numbers whose factorial contains the same number of digits as the number itself. That is to say n = digits_in(n!).
The trivial case is obviously...
Using log identities:
##log((\alpha - 1)!^2) = 2(log(\alpha - 1)!)##
Then apply Stirling's Approximation
##(2[(\alpha - 1)log(\alpha - 1) - (\alpha - 1)##
## = 2(\alpha -1)log(\alpha -1) - 2\alpha+2##
Is this correct? I can't find a way to check this computationally.
This alternating series indentity with ascending and descending reciprocal factorials has me stumped.
\frac{1}{k! \, n!} + \frac{-1}{(k+1)! \, (n-1)!} + \frac{1}{(k+2)! \, (n-2)!} \cdots \frac{(-1)^n}{(k+n)! \, (0)!} = \frac{1}{(k-1)! \, n! \, (k+n)}
Or more compactly,
\sum_{r=0}^{n} (...
The double factorial, ##n!## (not to be confused with ##(n!)!##), can be defined for positive integer values like so:
$$n!=n(n−2)(n−4)(n−6)...(n-a)$$
Where ##(n−a)=1## if ##n## is odd or ##(n−a)=2## if ##n## is even. Additionally, the definition of the double factorial extends such that...
Sometimes there are functions that are initially defined for only integer values of the argument, but can be extended to functions of real variable by some obvious way. An example of this is the factorial ##n!## which is extended to a gamma function by a convenient integral definition.
So, if I...
Hello.
I am trying to decipher the formula, making sure I understand what exactly is going on in each part of the expression. I will be grateful for your guidance, corrections and help.
Below I show the formula and the example for only 3 possible outcomes (in general it would be k)
\(p =...
Hello!
In this webpage:
https://onlinecourses.science.psu.edu/stat503/node/35
it describes the 2^k factorial experiment design. I understand that k is the number of factors that we are investigating (in this case two, a and b), 2 are the levels of each factor (+/-) and 2^k=4 is the number of...
Homework Statement
Prove that for a positive integer, p:
https://www.physicsforums.com/posts/5859454/I've tried this to little avail for the better part of an hour - I know there's a double factorial somewhere down the line but I've been unable to expand for the correct expression in terms of...
I understand that the standard proof is a bit different from my own, but I want to know if my reasoning is valid. PROOF:
Firstly, I assume that x is positive.
I then consider p = inf{n∈ℕ : n>x} . In other words, I choose "p" to be the smallest natural number greater than x. If we choose n>p...
In how many different ways can we arrange three letters A, B, and C? There are three candidates for the third position that leaves the two remaining letters for the second position and so 3 times 2 is 6 and One is the multiplicative identity I am astonished by The commutative property of...
Homework Statement
What is the value of ## \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + ... ## ?
Homework Equations
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I have no idea since it's neither a geometric nor arithmatic seriesThe Attempt at a Solution
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My Calculus purcell book tells me that it is e - 1 ≈...
Homework Statement
Determine whether the series converges or diverges.
∞
∑ 1/n!
n=1
Homework Equations
If ∑bn is convergent and an≤bn for all n, then ∑an is also convergent.
Suppose that ∑an and ∑bn are series with positive terms. If
lim an = C
n→∞ bn
where c is finite number and c>o...
Homework Statement
Hello all,
I am trying to determine the last hexadecimal digit of a sum of rather large factorials. To start, I have the sum 990! + 991! +...+1000!. I am trying to find the last hex digit of a larger sum than this, but I think all I need is a push in the right direction...
We are starting sequences, and in one of the examples we have this limit:
$$\lim_{{n}\to{\infty}} \frac{R^n}{n!}$$
We let $M$ equal a non-negative integer such that $ M \le R < M + 1$
I don't get the following step:
For $n > M$, we write $Rn/n!$ as a product of n factors:
$$\frac{R^n}{n!}...
Is there a way to identify a factorial without referring to computation of a factorial?
For example, is there a way to identify 5040 as a factorial and a way to identify 5050 as not a factorial?
I am trying to work through a simplication of this factorial with variables:
(n/2)!/[(n+2)/2]!
I get,
2[n(n-1)]/2[(n+2)(n+1)n(n-1)]
cancelling the 2[n(n-1)]
leaves me with 1/[(n+2)(n+1)]
However, Wolfram Alpha tells me this can be simplified as 2/(n+2) and I don't see that.
Thanks
Homework Statement
I have to determine whether or not the following sequence is convergent, and if it is convergent, I have to find the limit.
an = (-2)n / (n!)
In solving this problem, I am not allowed to use any form or variation of the Ratio Test.
2. The attempt at a solution
I was...
Hi guys, my english is very bad but let me try translate the question:
Let $ a_n=\frac{(n+9)!}{(n-1)!} $ . Let k the lesser natural number since that the first digit (on the right side) after all the zeros of $ a_k $ is odd.
Example: $ a_k =$ 4230000000 or $ a_k =$ 62345000
This odd digit...
Simplify. Assume that $n$ and $m$ are positive integers,
$a>b$, and $a>2$.$\frac{\left(a+1\right)!}{\left(a-2\right)!}$
was helping a friend with this but was clueless
I know that n! $=n(n-1)(n-2) ... $
hey so
if you are taking a floor function of a fraction >1, is there any way to predict anything about it's factorization?
what about when the numerator is a factorial and the denominator is made up of factors that divide said factorial but to larger exponents then those that divide the...
Hi, question is - show that the following series is convergent: $ \sum_{s}^{} \frac{(2s-1)!}{(2s)!(2s+1)}$
Hint: Stirlings asymptotic formula - which I find is : $n! = \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n $
I can see how this formula would simplify - but can't see how it relates to the...
A few decades ago my algebra teacher showed how to construct the expression for binomial coefficients. If I start with Pascal's recursion, and propose C(n,k)=n!/k!(n-k)!, I can prove it to be so through induction. But that doesn't give me that happy feeling that comes with understanding.
It...
number 15 questions b and c are giving me a very hard time. I have tried expanding them then factoring out the common terms but somehow not getting it to be proven. detailed help will be appreciated.
program factorial
implicit none
integer::fact,i,n
print*,'enter the value of n'
read*,n
fact=1
do i=1,n
fact=fact*i
end do
print*,'factorial is ',fact
end program
when input n largest number then answer is incorrect. how to solve
How can I find out if 1/n! is divergent or convergent?
I cannot solve it using integral test because the expression contains a factorial.
I also tried solving it using Divergence test. The limit of 1/n! as n approaches infinity is zero. So it follows that no information can be obtained using...
Ok, I had never seen a factorial problem like this, and the answer(n=7) didn't help me much in understand the solution either.
If (n+1)!/(n-1)! = 56 , what's the value of n?
Homework Statement
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I have to find the radius of convergence and convergence interval. So for what x's the series converge.
The answer is supposed to be for every real number. So the interval is: (-∞, ∞).
So that must mean that the limit L = 0. So the radius of convergence [ which is...
Have I completed this 2 x 3 factorial ANOVA chart correctly?
A sample of ¬N = 36 is recruited to participate in a study about speech errors. The researchers believe that there is an interaction between whether the speaker is distracted (distracted or not distracted) and the difficulty of the...
Hey! :o
I am asked to write a RAM (Random Access Machines) program to compute $n!$ given input $n$.
Could you give me some hints how I could do that?? (Wondering)
Homework Statement
Solve the following limit:
$$ \lim_{n\rightarrow \infty }n\cdot\left ( \frac{2\cdot4\cdot6 \cdots (2n-2)}{1\cdot3\cdot5\cdots (2n-1)} \right )^{2}$$
The Attempt at a Solution
I don't know where to begin. Until know I've encountered limits which I could deal with in some way...
I am learning about statistical design of experiments, and in the process of mathematically rigorizing the concepts behind fractional factorial designs of resolution III, I derived an interesting equation:
$$k = \sum_{i=1}^{3}{\lceil{\log_2{k}}\rceil \choose i},$$ for which the solutions $k$...
My first question is: is this formula (at the bottom) a known formula?
In this subject i haven't explained how i build up the formula.
So far i think it is equal to the gamma function of Euler with
\Gamma\left(\frac{m_1}{m_2}+1\right)= \frac{m_1}{m_2}\ !
with
m_1 , m_2 \in...
Our integral
\int\limits_0^{\pi/2} \sin^{2a+1}(x)\,dx
Has a Factorial Form:
{(2^a a!)}^2 \over (2a+1)!
What is the process behind going from that integral to that factorial form?
My approach which is not very insightful:
I used mathematica to calculate the integral to return...
I'm watching V. Balakrishnan's video lectures on Classical Physics, and right now he's going through statistical mechanics.
In that regards he's talking about Stirlings formula, and at one point, he wrote an integral definition of the factorial like the following
n! =...
HelloI wish to prove that
\[ \forall\;n\in \mathbb{N}\; n! \leqslant n^n \]
First we let \(n\) be arbitrary. Now I first write \( n! \) as \( n\cdot(n-1)\cdot(n-2)\cdots 3\cdot 2\cdot 1\).
Now we see that
\[ n \geqslant (n-1)\;; n \geqslant (n-2)\;\ldots ;n \geqslant n- (n-1) \]
So we get
\[...