# Tagged Questions

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### Why is $\lim\limits_{n \to \infty} \frac{1}{n} \sqrt[n] {n^n}=1$ where $\lim\limits_{n \to \infty} \sqrt[n] {n!}=\infty$?

Why is $\lim\limits_{n \to \infty}\frac{1}{n} \sqrt[n] {n^n}=1$ where $\lim\limits_{n \to \infty} \sqrt[n] {n!}=\infty$ ? We all know that $n^n > n! \ : \forall n$ so how come the factorial ...
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### does this series converge / diverge

does this series converge/diverge conditionally or absolutly $\sum_{n=2}^{\infty} (-1)^n \cdot \frac{\sqrt{n}}{(-1)^n + \sqrt{n}} \cdot \sin(\frac{1}{\sqrt{n}})$ i can use the facts that: ...
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### Finding the limit of $\left(\frac{n}{n+1}\right)^n$

Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$ I'm pretty sure it goes to zero since $(n+1)^n > n^n$ but when I input large numbers it goes to $0.36$. Also, when ...
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### Prove that $\limsup\left(b_{n}a_{n}\right)=\limsup\left(a_{n}\right)$ when $\lim_{n\rightarrow\infty}\left(b_{n}\right)=1$

I'm having trouble with this homework question: "Let there be a sequence $\left(b_{n}\right)_{n=1}^{\infty}$ such that $\lim_{n\rightarrow\infty}\left(b_{n}\right)=1$. Let there also be some ...
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### evaluating a complex limit to the power of one third [on hold]

Evaluate lim as $x$ approaches $\infty,$ $\displaystyle[(x^3+x^2)^\frac13 - (x^3-x^2)^\frac13$.
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### Stuck on Infinite L'hopitals

I have been trying forever to figure out this problem, but I seem to get stuck in an infinite L'hopitals loop. See the question below: Find the value of the positive constant c such that: ...
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### I need some basic introduction to limits

So, I know you can obviously cut out a value if it is multiplying and dividing something at the same time, right? Like: $$\frac{4h-2xh-h^2}{h} = \frac{h(4-2x-h)}{h} = 4-2x-h$$ But then I saw this ...
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### Prove: $\sum {{a_{{n_k}}}} < \infty \Rightarrow \sum {|{a_n}| < \infty }$

Prove: $$\sum {{a_{{n_k}}}} < \infty \Rightarrow \sum {|{a_n}| < \infty }$$ In words, if every sub-series of $\sum a_n$ converges then $\sum a_n$ converges absolutely. I know that: ...
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### uniform convergence- sequence of functions

How can I prove that $f_n(x)=2/\sqrt n$ is uniformly convergent to cero in the interval $x \in (0, \infty)$? It obviously the sequence goes to cero but I would like to bound the limit with another ...
I've stumbled across this problem:$$\lim_{n\to\infty}\sqrt{n}(\sqrt[n]{3}-\sqrt[n]{2})$$ Intuition tells me that the square root infinity is weaker than the nth root zero, also ...