# Tagged Questions

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### 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
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### 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*} ...
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### 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 ...
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### Series — Coefficient Cn and Radius of Convergence

. I'm lost, and my textbook is failing me
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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: ...
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### 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 ...
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### Limit of the sequence $\frac{(n!)^{1/n}}{n}$ [duplicate]

Which is the limit of the fllowing sequence $$\frac{(n!)^{1/n}}{n}$$
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### 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.
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### 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?
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### 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 ...
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### Series involving factorials

How would one go about proving $$\int_{0}^1\frac{e^x-1}{x/2}\ dx=\sum_{n=0}^\infty\frac{1}{\binom{n+2}{2}}\frac{1}{n!}(0!+1!+2!+3!+...+n!)$$
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### 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 ?
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### 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 ...
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### 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. ...
Evaluate : $$\lim_{n\to \infty }\frac{n!}{{{n}^{n}}}\left( \sum\limits_{k=0}^{n}{\frac{{{n}^{k}}}{k!}-\sum\limits_{k=n+1}^{\infty }{\frac{{{n}^{k}}}{k!}}} \right)$$