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## A term for a unimodal function similar to $\exp(-x^2)$

Is there a term for a smooth $\mathbb R\to\mathbb R^+$ function that is unimodal (has one local maximum), and whose $n^{\text{th}}$ derivative has $(n+1)$ local extrema? An example of such a function is $\exp(-x^2)$. An example of a unimodal function that does not fit these criteria is $\frac1{1+x^4}$.

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## Does there exist an infinite series which takes some positive integer $n$ as a parameter and converges to the sum of sequence of cubes up to $n^3$?

Does there exist an infinite series with sequence {$a_k$}$\implies f(n,k)$ s.t for some positive integer $n$: $$\sum \limits_{k=0}^{\infty}f(n,k) = \frac{(n)^2(n+1)^2}{4}$$the right hand side being the sum of sequence of cubes for some positive integer $n$. For instance: $$\sum \limits_{k=0}^{\infty}f(2,k) = 9$$ $$\sum \limits_{k=0}^{\infty}f(3,k) = 36$$

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## How do i prove that $\lim_{x\rightarrow 4}⁡ (\sqrt x-2 )= 0$ with the definition? [closed]

I have no idea how to prove that $$\lim_{x\rightarrow 4}⁡ (\sqrt x-2 )= 0$$ using $\delta$ and $\varepsilon$ My attempt after the hint ( Please, I barely know what I’m doing this could all be gibberish and I wouldn’t be any wiser): $0 < (x-4) < \delta$ If $|x-4| < 1$ then $x < 5$ […]

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## Disproving Multivariable Limit

I just started learning about multivariable functions, and I was wondering if there is something like Heine for disproving the limit of multivariable functions. For example: I know that we can approach multivariable functions from different directions to see if the function keep the same value, but is this enough? Is there a more rigorous […]