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

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0
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1answer
21 views

Real roots for exponential-polynomial equations

I am trying to find the number of real roots of an equation such as $k_1 x e^x-e^{k_2 x}-k_3x+k_4=0.$ Setting the first derivative equals to zero is analytically unsolvable, unfortunately. Do you ...
1
vote
1answer
26 views

Is $e^{im\phi}$ equivalent to $e^{ix}$?

In this spherical harmonics paper, there's a periodic function $\Phi(\phi)$ defined as $$\Phi(\phi)= \bigg\{ \begin{array} \\e^{im\phi} \\e^{-im\phi} \end{array} \space m = 0,1,2,3...$$ Is $e^{im\...
-1
votes
1answer
77 views

How to find the function such that $\int_0^1f(x)\ \mathrm dx=e^{-4n^{2}{\pi}}$ [on hold]

Find $f(x)$ where: $$ \int_{0}^{1}f(x,n)\ \mathrm dx=e^{-4n^{2}{\pi}} $$ Is it possible that question contains infinitely many answers? How to solve this ? Please provide me a hint.
2
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0answers
27 views

Maxima of $f(x)/e^x$ where $f(x)$ is an approximation of $e^x$ using Stirling's

Let $$f(x)=1+\sum_{n=1}^\infty\frac{x^n}{\sqrt{2\pi n}(n/e)^n}\tag1$$ and let $$g(x)=\frac{f(x)}{e^x}\tag2$$ If we plot $g(x)$ we get a graph that looks like this: Clearly there is a maximum at ...
0
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3answers
56 views

Solve $3^{2x}-3^x\geq2$

How to solve: $$3^{2x}-3^x\geq2$$ I tried with $y=3^x$ and solved as equation: $y^2-y-2 \geq 0$ and I get: $y<2$ $y>-1$ How should I proceed?
1
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2answers
34 views

If the radioactive isotope strontium $240$ has a half life of $120$ years, how long until it decays to only $60\%$ of its original radioactivity?

I been trying to solve this problem for hours and the only thing i came up with, was a formula for their relationship. $1/2A = A_0 e^{120r}$ $\ln(1/2 = e^{120}r)$ $\ln(1/2) = 120r$ $r = \ln(0.5)/...
0
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2answers
50 views

Power series as approximation

I have to estimate the error when I approximate the function $$e^{\sin x}$$ to $$1+x+x^{2}+x^{3}$$ when $|x|<0.1$. I really don't know how to do because my teacher didn't teach me. But what I did ...
2
votes
3answers
26 views

Exponential limit convergence for each $x$

I have $f_n(x)=\left( 1+\frac{-e^{-x}}{n} \right)^n$, what about the convergence to $f(x)=e^{-e^{-x}}$? is it true $\forall x?$ I say yes, but how can I show this? Is continuity of $f_n$ enough?
26
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15answers
2k views

Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$

At this link someone asked how to prove rigorously that $$ \lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x. $$ What good intuitive arguments exist for this statement? Later edit: . .&...
0
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3answers
44 views

How to determine Laurent series associated to $f(z)$ [on hold]

The function is $$f(z)= \frac{1}{(e^z -1)},$$ $z$ belong to $\mathbb{C}$ and $0<|z|<1$. I need a general expression in term of a sum from 0 to infinity
0
votes
2answers
25 views

Value of $\lambda$

If $y=e^{2\cos^{-1}x}$ also $$(1-x^2)y_2-xy_1-\lambda y=0$$ then the value of $\lambda$ is. I see that the question is incomplete but answer is given as $2$. Am I missing on anything.
3
votes
4answers
101 views

Solve $2^x+4^x=2$

This is the equation, but the result is different from wolframalpha: $$2^x+4^x=2$$ $$2^x+2^{2x}=2^1$$ $$x+2x=1$$ $$x=\frac{1}{3}$$ WolframAlpha: $x=0$ Where is the error?
0
votes
1answer
35 views

Limits involving $e$ [on hold]

I am looking to solve the following limits. $\displaystyle\lim_{x\to-\infty}5e^{-x}$ and $\displaystyle\lim_{x\to2}\frac{1}{2e-ex}$ Any help would be appreciated.
9
votes
1answer
100 views

Fake proof that $\frac{e^x-1}{e^x+1}=e^x$, via integrating $\operatorname{sech} x$ in two ways

We start with the integral: $$\int \text{sech}(x)dx$$ Method 1 \begin{align} \int \text{sech}(x)dx & = \int\frac{2}{e^x+e^{-x}}dx \\ &= \int\frac{2e^x}{e^{2x}+1}dx \end{align} Using the ...
0
votes
2answers
44 views

Prove that $\frac{1}{2} (e^R - e^{-R}) \geqslant \frac{1}{4} e^R $ with $R > 0$

Sorry to bother you with silly question, but I can't figure out how to prove: $$\frac{1}{2} (e^R - e^{-R}) \geqslant \frac{1}{4} e^R $$ with $R > 0$. I tried different ways but that didn't lead ...
2
votes
6answers
132 views

Solve equations like $3^x+4^x=7^x$

How can I solve something like this? $$3^x+4^x=7^x$$ I know that $x=1$, but I don't know how to find it. Thank you!
2
votes
3answers
42 views

Identity with exponential function: $\lim_{n\to\infty}\frac{n^{2n}}{(n+1)^{2n}} = \frac{1}{e^2}$

Could you please explain me how we got this identity $\lim_{n\rightarrow \infty}\frac{n^{2n}}{(n+1)^{2n}} = \frac{1}{e^2}$ when we know $\lim_{n\rightarrow \infty}(1+\frac{1}{n})^n = e$ Thanks!
0
votes
1answer
22 views

applying exponents in an absolute value brackets

So part of the problem I'm trying to solve is this: $|2-9|^{3{^3}}$ being the exponent, to the power of 3. Do I have to apply the exponent to everything in the bracket? 2*2*2 and 9*9*9 or does the ...
3
votes
0answers
67 views

Does this equation have no solutions?

The question is this : The source from where I got this question was devoid of any answers to it, so I came here, this is how I proceeded : LHS : $((((({(x)^x})^{2x})^{3x})^{....x^2})^2 = (((((x)^...
0
votes
1answer
429 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the $y$...
0
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3answers
42 views

Does this sequence have a closed form representation?

We know that $$ \sum_{s=0}^\infty \frac{\lambda^{s}}{s!} = e^\lambda$$ Relatedly, $$ \sum_{s=1}^\infty \frac{\lambda^{s}}{s!}s = \lambda \sum_{s=1}^\infty \frac{\lambda^{s-1}}{(s-1)!}$$ For which ...
3
votes
1answer
61 views

Proving that the exponential function is continuous

We aren't allowed to use many tricks such as difference quotient / integral calculus... Prove that $\exp$ is continuous at $x_{0}=0$ .....................................................................
2
votes
1answer
77 views

Is it valid to write $1 = \lim_{x \rightarrow 0} \frac{e^x-1}{x} = \frac{\lim_{x \rightarrow 0} (e^x) -1}{\lim_{x \rightarrow 0} x}$?

I just want to clarify one thing I was never really sure on. First the question: $$1 = \lim_{x \rightarrow 0} \frac{e^x-1}{x} = \frac{\lim_{x \rightarrow 0} (e^x) -1}{\lim_{x \rightarrow 0} x}$$ is ...
0
votes
2answers
101 views

Proving that $\pi$ and $e$ are rational numbers [duplicate]

Maybe this question is too dumb to be asked, but it's really bugging me so I decide to ask it anyway. I hope you bear with me. Okay, it's known that both sides of the following series equal. $$\pi=...
4
votes
1answer
70 views

Ways to squeeze $e$ by hand

Let $a$ and $b$ be the lower and upper bound of $e$, respectively. Both $a$ and $b$ are rational numbers. Without using a calculator and without knowing the value of $e$, find $a$ and $b$ where $b-a&...
7
votes
3answers
140 views

Proving $\pi \gt e+\frac{1}{e} \gt \pi-\frac{1}{\pi} \gt e$

I created this problem for myself as a fun exercise. I want to prove the following statement: $$\pi \gt e+\dfrac{1}{e} \gt \pi-\dfrac{1}{\pi} \gt e$$ I found that the following upper/lower bounds ...
0
votes
3answers
51 views

how so simplify this exponential equations

((a^3/2)/(b^3))/((a^-1)/(b^2)) I tried to solve this problem many times, however I tend to get the wrong answer. Here is the method I tried (((a^3)^1/2)/(b^3))*... sorry I get confused i got (...
0
votes
2answers
71 views

How can I get the expression of x?

If there is $$x^2e^{A\sqrt x}=B$$ then what is the expression of $x$? If this cannot be solved, is there any approximation?
4
votes
2answers
71 views

Proofing that the exponential function is continuous in every $x_{0}$

Given: $$\exp: \mathbb{R} \ni x \mapsto \sum_{k=0}^{\infty } \frac{1}{k!} x^{k} \in \mathbb{R}$$ also $e = \exp(1)$. For all $x \in \mathbb{R}$ with $\left | x \right | \leq 1$: $$\left | \exp(x) -...
0
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0answers
27 views

Integrate this expression containing exponential

I am trying to integrate $$ \int x^a f'(x)\lambda \exp(f(x)\lambda) dx$$ I'm not that great at these things, but I noticed that $f'(x)\lambda \exp(f(x)\lambda = \frac{d}{dx} \exp(f(x)\lambda)$. I ...
3
votes
1answer
57 views

Calculate: $f(a)=\int\limits_{-\infty}^{\infty} \exp(-|x|^a)\mathrm{d}x$

Given the following function: $$ f(a)=\int_{-\infty}^{\infty} \exp(-|x|^a)\mathrm{d}x $$ For which values of $a$ is it possible to give an exact value for this function? I only know $f(2)=\sqrt{\pi}$...
1
vote
1answer
29 views

How can I prove this inequality involving the exponential function?

Given $$\exp: \mathbb{R} \ni x \mapsto \sum_{k=0}^{\infty } \frac{1}{k!} x^{k} \in \mathbb{R}$$ also $e = \exp(1)$. For all $x \in \mathbb{R}$ with $\left | x \right | \leq 1$: $$\left | \exp(x) - ...
0
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1answer
29 views
0
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0answers
7 views

How for continuous compounding of interest, the difference in balance over difference in time is equal to interest rate times balance at the instant?

I was reading the chapter on exponential growth and decay in Morris Klein's Calculus book. He says $\delta A=A(0.04)\delta t$, 0.04 the interest is in unit percent per year. A is balance. t is time. ...
0
votes
0answers
20 views

Determining bounds for a sum with nested infinite series

I am computing the inner product of the characters of the trivial and the $k$-th irreducible two dimensional representations of the dihedral group $D_n$ of order $2 n$ when $n$ is even. The ...
-1
votes
1answer
89 views

Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$ [closed]

I'm not sure how to go about this. Does it involve using MVT? I got as far as saying $e = \frac{e^x - e^y}{x-y}$.
3
votes
2answers
153 views

Derivative of the matrix exponential with respect to its matrix argument

I was trying to find the Frechet derivative of $f = \exp(X)$, where $X \in \mathbb{R}^{n\times n}$ is positive definite. I thought it ought to be $\exp(X)$. I see results where the derivative is with ...
3
votes
2answers
86 views

I can't complete the integration of $e^{\sqrt{x}}$

Compute $\displaystyle\int_0^1e^{\sqrt{x}}\,dx$ That's a picture of how far I could get while trying to integrate $e^{\sqrt{x}}$. I tried the substitution method first, (boxed part) and then went ...
6
votes
7answers
218 views

Proof of Euler's formula that doesn't use differentiation?

So I saw a 'proof' of the sine and cosine angle addition formulae, i.e. $\sin(x+y)=\sin x\cos y+\cos x \sin y$, using Euler's formula, $e^{ix}=\cos x+i\sin x$. By multiplying by $e^{iy}$, you can get ...
0
votes
1answer
22 views

Exponential decay + a recurrence relation

I'm not sure if I get this right, some pointers could be helpful. Say you have to take 60m of some sort of medication at midnight. It has a blood half-life of 6 hours. Meaning that after 24 hours 3....
0
votes
1answer
92 views

What is the value of $e^{-10000}$?

What is the value of $e^{-10000}$? We know that the function $e$ does not attain value $0$ anymore. But in R and Matlab the value of $e^{-10000}$ is given as $0$ which is not correct anymore. I ...
0
votes
1answer
19 views

Bounding an exponential integral

I'm having trouble seeing this bound I've seen on a proof. Let $f$ be a polynomial, and $F$ the polynomial obtained from $f$ by replacing each coefficient by its absolute value. Then: $$\bigg{|}\...
0
votes
0answers
39 views

Prove Exponential series from Binomial Expansion

I try to prove the Exponential series : $$\exp(x) = \sum_{k=0}^{\infty} \dfrac{x^k}{k!}$$ From the definition of the exponential function $$\exp(x) \stackrel{\mathrm{def}}{=} \lim_{n\to\infty} \left(...
37
votes
1answer
691 views

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ $$u_2=v_2=...
3
votes
3answers
218 views

Lambert W function?

I need to solve the next equation: $$ax^{bx+c}=d$$ where a, b, c and d are positive real values. Do I need to use Lambert W function, or there is some other method? Thanks!
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votes
2answers
64 views

Exponential of a number [closed]

What is exponential of a number? $E^{10}=22026.4657948$ What is the mathematical calculation behind the above calculation? Regards, Philip
8
votes
6answers
276 views

Are the any non-trivial functions where $f(x)=f'(x)$ not of the form $Ae^x$

This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is ...
0
votes
1answer
40 views

Proof that $a^x$ goes towards infinity as x goes towards infinity

I'm tasked to prove that $a^x \rightarrow \infty $ when $x \rightarrow \infty$ provided that (a > 1). I've found a very rigorous proof for this. But my question is, why can't it be logically realized ...
59
votes
13answers
42k views
1
vote
4answers
80 views

Prove that for all $a > 0$: $\int_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \frac{\pi}{2} - \int_0^a\frac{\sin x}{x}dx$

Prove that for all $a > 0$: $$\int\limits_0^{\pi/2}e^{-a\cos x}\cos(a\sin x)dx = \cfrac{\pi}{2} - \int\limits_0^a\cfrac{\sin x}{x}dx$$ I have no idea how to solve it. But the task looks very ...