Tagged Questions
0
votes
1answer
104 views
Derivative of $\frac{e^x}{x!}$
I am having a bit of trouble putting all the differentiation rules together with the following problem:
$$ \frac{d}{da} \Bigg(\frac{a^x}{x!}e^{-a}\Bigg)$$
Where $x$ is a discrete variable and $a$ is ...
18
votes
4answers
357 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. ...
8
votes
4answers
585 views
Does $a!b!$ always divide $(a+b)!$
Hello the question is as stated above and is given to us in the context of group theory, specifically under the heading of isomorphism and products. I would write down what I have tried so far but I ...
0
votes
1answer
63 views
Colored Blocks Factorial
I was given the following problem:
A bag contains 14 red blocks, 10 white blocks and 12 blue blocks. How many different 19 block sets can be created from the collection in the bag? Two block sets ...
3
votes
1answer
72 views
Factorial Identity - True or False?
Let $x$ and $y$ be positive integers.
Then, is
\begin{align}
\frac{x^{xy}}{(xy)!} = \sum_{k_1+...+k_x = xy} \frac{1}{(k_1)!...(k_x)!}
\end{align}
true, where $k_1$, ..., $k_x$ are all positive ...
8
votes
6answers
443 views
Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$
Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$.
3
votes
1answer
441 views
How to prove that $\mathrm{Fibonacci}(n) \leq n!$, for $n\geq 0$
I am trying to prove it by induction, but I'm stuck
$$\mathrm{fib}(0) = 0 < 0! = 1;$$
$$\mathrm{fib}(1) = 1 = 1! = 1;$$
Base case n = 2,
$$\mathrm{fib}(2) = 1 < 2! = 2;$$
Inductive case ...
3
votes
1answer
256 views
Smallest positive integer
Let S(m) is the sum of the factorials of the digits of integer m.
I try to find the smallest positive integer n with S(n)=111.
My answer is 12334444. Is it right?
1
vote
1answer
88 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: ...
7
votes
2answers
214 views
Finding all the numbers that fit $x! + y! = z!$
I have the formula $x! + y! = z!$ and I'm looking for positive integers that make it true. Upon inspection it seems that x = y = 1 and z = 2 is the only solution. The problem is how to show it.
...
3
votes
0answers
152 views
How to find $\beta$ and $\alpha$?
$\mathbb{P}$ is the prime numbers set.
$p \in \mathbb{P}$
$a,b,c \in \mathbb{N}$
$n=a p^b+c$ where
$c= n\bmod p$
$b$ is the highest power of $p$ who divides $n-c$
How to find $\beta$ where ...
0
votes
2answers
353 views
Simplifying this factorial expression
I know that $\frac{(m-1)!}{(m-n)!(n-1)!} + \frac{(m-1)!}{(m-n-1)!(n)!} = \frac{m!}{(n)!(m-n)!}$, but I am not sure on the intermediate steps. The only solution I am seeing involves finding a common ...
4
votes
2answers
811 views
How to prove $a^n < n!$ for all $n$ sufficiently large, and $n! \leq n^n$ for all $n$, by induction?
I have a couple things I want to prove. I'm pretty sure a proof by induction is the best route for these.
First, I need to show that $5^n < n!$ from some $n_{0} > 0$. I'm choosing $n_{0} = ...
