# Tagged Questions

For question involving exponential functions and questions on exponential growth or decay.

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### Finding the limit of the sequence $x(n) = (1+2/n)^n$ [duplicate]

What we are allowed to use - 1) The fact that limit of $(1+1/n)^n$ exists and assumed to be some real number $e$ 2) Subsequencial properties of limits of sequences 3) Basic properties of limit In the ...
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### Solve equation with exponentials

I'm trying to obtain $x$ in the following equation: $$1= ae^{bx}+ce^{dx}$$ with a,b,c,d known. What I did was to take the ln of all the equation, so I have: $$\log\frac{1/ac}{b+d} = x$$ ...
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### General formula for the higher order derivatives of composition with exponential function

Suppose I have a function $x:\mathbb{R} \to \mathbb{R}$ and consider: $$g(t) = e^{x(t)}$$ When I start differentiating with respect to $t$ I obtain: \begin{align} g'&=e^xx'\\ g''&=e^x((x')^2+x'...
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### How to prove the equality or inequality

Can anybody prove that the following equation is right or wrong? $$\int_0^te^{-t}(1-e^{-2x})^ke^x dx=\int_0^t2k(e^{-2x}-e^{-t-x})(1-e^{-2x})^{k-1}dx$$ where $t>0$ and $k$ is and integer. My small ...
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### Proof that $|1 - e^{i \theta}| \geq \frac{2|\theta|}{\pi}$ for $-\pi \leq \theta \leq \pi$?

I would like to prove (geometrically if possible) the above result. Could someone help?
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### Why is $\frac{(e^x+e^{-x})}{2}$ less than $e^\frac{x^2}{2}$?

I have read somewhere that this equality holds for all $x \in \mathbb {R}$. Is it true, and if so, why is that? $$\frac{(e^x+e^{-x})}{2} \leq e^\frac{x^2}{2}$$
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### Find the limit of the sequence $(1-1/n)^n$

All that we have proven so far is that limit $(1+1/n)^n$ exists and considered to be a number 'e' which belongs to $(2,3)$ We haven't proven that 'e' is irrational or that lim $(1+(x/n))^n) = e^x$ ...
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### exponentiating a matrix and sum of elements

$$M= \begin{bmatrix} 1&1&0\\0&1&1\\0&0&1\\ \end{bmatrix}$$ Then the sum of all entries of $e^{M}$ i just don't know how to calculate this sum as this would be an infinite ...
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### Generalizing exponential moving average to n samples

Assume that we have a moving average like this: $E_t = a*S_{t-1}+(1-a)*E_{t-1}$ where $E$ would be an estimate we are interested in, and $S$ is a sample we take at each point in time. Now, if we ...
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### Supremum of $\cot(\pi z)$ where $z$ is on circle with radius $n+1/2$

I try to estimate the supremum of $|\cot(\pi z)|$ and where $z=(n+1/2) e^{i t}$, $n\in\mathbb N$ and $t\in[0,2\pi)$. I should be a constant. So far I did by wiriting it in exponential form and ...
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### Why does $\frac{1}{6e^{2y}}=\frac{1}{2x-8}$ in this context?

This is the context: I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
### Integration of $\frac{x^2}{2\left(e^x+1\right)}$
Let: $$f(x) = \int \frac{x^2}{2\left(e^x+1\right)}dx$$ Is there a way to find $f(x)$? I've tried through integration by parts, but that didn't work out. If substitution is the answer, I can't see ...