Questions tagged [exponential-function]

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

7
votes
3answers
1k views

Limit involving $(\sin x) /x -\cos x $ and $(e^{2x}-1)/(2x)$, without l'Hôpital

Find: $$\lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)}$$ I have factorized it in this manner in an attempt to use the formulae. I have tried to use that ...
6
votes
3answers
245 views

How to prove $(\frac{n+1}{e})^n<n!<e(\frac{n+1}{e})^{n+1}$ without integrating method?

How to prove $$\left(\frac{n+1}{e}\right)^n<n!<e\left(\frac{n+1}{e}\right)^{n+1}$$ without integrating method? In fact we could prove this by noticing that $$i<x<i+1\Rightarrow \ln i<\...
6
votes
3answers
385 views

Solving $x^{x^x}=c$ with Lambert's W function

We can solve the equation $x^x=c$ by the use of Lambert W function but how do we solve $x^{x^x}=c$ ,
5
votes
3answers
227 views

What is $ \lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$?

How to solve the following limit question? $$\lim_{n\to\infty}\frac{1}{e^n}\Bigl(1+\frac1n\Bigr)^{n^2}$$ Thanks a lot.
4
votes
3answers
205 views

Particular definition of e

Show that, $$e=\lim_{x\to \infty} \left(1+\frac{1}{x}\right)^x $$ Is the same number that satisfies, $$\lim_{h\to0} \frac{e^{h}-1}{h} = 1 \tag{*}$$ You don't have to do that if it's too cumbersome, ...
4
votes
5answers
411 views

Why does the Taylor expansion of $e^x$ satisfy exponential properties?

Suppose I knew nothing about the function $e^x$. If I wanted to find a power series that was its own derivative, I would logically start with the constant term, and first start by setting it to $1$. ...
3
votes
4answers
1k views

Quick way to solve the system $\displaystyle \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} = \frac{65}{36}$, $xy-x+y=118$.

Consider the system $$\begin{aligned} \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} & = \frac{65}{36}, \\ xy -x +y & = 118. \end{aligned}$$ I have solved it by ...
3
votes
6answers
224 views

Is there an analytic solution for the equation $\log_{2}{x}+\log_{3}{x}+\log_{4}{x}=1$?

I am looking for a close form solution for below equation. $$\log_{2}{x}+\log_{3}{x}+\log_{4}{x}=1.$$ I solve it by graphing, but I don't know is there a way to find $x$ analytically ?
2
votes
3answers
807 views

The matrix exponential: Any good books?

Looking for a book/article with a lucid exposition of the matrix exponential, preferably including the case of infinite matrices. Basic properties especially, but also differential equations are of ...
1
vote
2answers
100 views

The least value of $b$ such that $2^{3^{\cdots^a}}\leq b^{(b-1)^{\cdots^2}}$

I'm interested in an upper bound for the minimal positive integer $b$ for which $$2^{3^{4^{\cdots^a}}}\leq b^{(b-1)^{\cdots^{3^{2}}}}$$ holds, given a positive integer $a\geq2$. If possible, but I ...
1
vote
0answers
295 views

Find the positive root of the equation $ce^{-c}-2(1-e^{-c})^2=0$

Can you help me find a root for $c$ in the equation below? $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ By expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, $$(c+4)e^{-c}-2-2e^{-2c}=0 ...
14
votes
4answers
13k views

How do pocket calculators calculate exponents?

I'd like to know specifically how a pocket calculator (TI calculators also apply) calculates $e^{0.1}$, and what methods or algorithms pocket calculators use in order to produce their answer.
12
votes
6answers
603 views

$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
8
votes
4answers
2k views

How unique is $e$?

Is the property of a function being its own derivative unique to $e^x$, or are there other functions with this property? My working for $e$ is that for any $y=a^x$, $ln(y)=x\ln a$, so $\frac{dy}{dx}=\...
7
votes
5answers
438 views

What is a nice way to compute $f(x) = x / (\exp(x) - 1)$?

I want it to be stable near $f(0) = 1$. Is there a nice function that does this already, like maybe a hyperbolic trig function or something like expm1, or should I just check if $x$ is near zero and ...
5
votes
3answers
140 views

Stuck on this integral involving exp and the floor function

Here is the integral $$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$ Here is what I have so far: $$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$ $$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$ $$ = \...
4
votes
3answers
303 views

Iterates of $f_b(x) = x - \log_b(x) $ - for $\log(b) \approx 0.399$: convergence to accumulation points or chaos?

In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm): $$ f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$ with the ...
4
votes
1answer
177 views

Exponential Diophantine: $2^{3x}+17=y^2$

Is there a way of solving the following equation, in integers $(x,y)$, by hand? : $2^{3x}+17=y^2$. You can also try: $2^{2x}+17=y^2$ or more generally $2^x+17=y^2$; each of these has at least 1 ...
4
votes
4answers
263 views

Matrix exponential of a traceless $2\times2$ complex matrix

How to calculate $\exp\left(t\begin{bmatrix}0 & z\\z^* & 0\end{bmatrix}\right)$, or $\exp\left(\begin{bmatrix}0 & v\\w & 0\end{bmatrix}\right)$ (where: $v, w \in \mathbb{C}$) in ...
4
votes
2answers
335 views

Is there any proof for this formula $\lim_{n \to ∞} \prod_{k=1}^n \left (1+\dfrac {kx}{n^2} \right) =e^{x⁄2}$

Some times ago, In a mathematical problem book I sow that this formula. I don't no whether it is true or not. But now I'm try to prove it. I have no idea how to begin it. Any hint or reference would ...
4
votes
4answers
3k views

The definition of e by limits of $(1+1/n)^n$ through series expansion

I think the problem I have is due to not being knowledgeable about limits. If I use binomial expansion to expand $(1+1/n)^n$ to $1 + \frac{n!}{(n-k)!k!}*(1/n)^k + ...$, I can imagine replacing $n$ ...
3
votes
1answer
302 views

integral evaluation of an exponential

let be the function $$ e^{-a|x|^{b}} $$ with $ a,b $ positive numbers bigger than zero then how could i evaluate this 2 integrals ? $$ \int_{-\infty}^{\infty}dxe^{-a|x|^{b}}e^{cx}$$ here 'c' can ...
3
votes
4answers
110 views

Prove by induction: $n! \ge 2^{(n-1)}$ for any $n \ge 1$ [duplicate]

I need help with this exercise. What I've done so far is prove the exercise when $n=1$. So: $$n=1$$ $$1!\ge2^{(1-1)}$$ $$1\ge2^0$$ $$1\ge1$$ Which is true Therefore, now that I assume that ...
2
votes
1answer
9k views

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 $(1+(x/n))^n) = e^x$ ...
2
votes
3answers
12k views

On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?

I was seeing the proof that $\exp(A)\exp(B)=\exp(A+B)$ on link Show that $ e^{A+B}=e^A e^B$ where uses the hypothesis $AB=BA$? Thanks!
1
vote
2answers
245 views

Generalisation of Lambert W function?

I want to solve an equation of the form: $\exp(C / x) - 1 = D / (x + a)$ This seems to be almost in a form where I can express solutions in terms of the Lambert W function but I can't seem to figure ...
0
votes
6answers
373 views

how to prove that $\lim\limits_{n\to\infty} \left( 1-\frac{1}{n} \right)^n = \frac{1}{e}$ [duplicate]

I need to prove $$\lim\limits_{n\to\infty} \left( 1-\frac{1}{n} \right)^n = \frac{1}{e}$$ For now, I only know the e definition: $\lim\limits_{n\to\infty} \left ( 1+\frac{1}{n} \right)^n = e $, and I ...
0
votes
2answers
172 views

Showing $(n+1)^n<e^nn!$ by induction

Show $(n+1)^n<e^nn!$ I know why that would be the case using general knowledge and a bit of substitution but am clueless on how to prove it.
9
votes
3answers
578 views

Proving that a definition of e is unique

We can define $e$ as the number such that $\lim_{h \to 0} \frac{e^h-1}{h}=1$. However, of course we can only define $e$ this way if it is unique, i.e., there is no other value $c$ for which that is a ...
8
votes
7answers
8k views

Prove $\exp(x+y) = \exp(x) \exp(y)$ for $\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$

I am trying to prove $\exp(x+y) = \exp(x) \exp(y)$. I may use that $$\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$$ I further know how to multiply two power series in one point, i.e. if $f(x) = \sum_{...
7
votes
5answers
4k views

How to integrate $\int \frac{e^x dx}{1\,+\,e^{2x}}$

Ok, I give up, I have tried with $u$-substitution and integration by parts but I can't solve it. The integral is: $$\int{\frac{e^x dx}{1+e^{2x}}}$$ I have tried $u=e^x$, $u=e^{2x}$ and also ...
4
votes
4answers
14k views

Which has a higher order of growth, n! or n^n? [duplicate]

In our algorithms class, my professor insists that n! has a higher order of growth than n^n. This doesn't make sense to me, when I work through what each expression means. ...
4
votes
2answers
120 views

Intuition for why equations of the form $k^x=x^c$ are not solvable trivially?

It recently occurred to me that I did not know how to solve equations of the form $k^x=x^c$ for any two constants $k$ and $c$. After much pain in algebraically manipulating the equation (using ...
4
votes
1answer
632 views

How did Bernoulli approximate $e$?

Researching on the internet, it is easy to find that Bernoulli was the first to give a one-digit approximation for $e$ (specifically, $2.5<e<3$). But, I cannot find any source describing ...
4
votes
2answers
124 views

Where we have used the condition that $ST=TS$, i.e, commutativity?

definition Let $A$ be an $n\times n$ matrix. Then for $t\in \mathbb R$, $$e^{At}=\sum_{k=0}^\infty \frac{A^kt^k}{k!}\tag{1}$$ Proposition If $S$ and $T$ are linear transformations on $\mathbb R^n$ ...
4
votes
3answers
330 views

Prove that if $\phi'(x) = \phi(x)$ and $\phi(0)=0$, then $\phi(x)\equiv 0$. Use this to prove the identity $e^{a+b} = e^a e^b$.

I am given the following. hint Consider $f(x)=e^{-x} \phi(x)$. I am unsure how to approach this problem.
3
votes
5answers
125 views

Show that $\cos(x) \le e^{-x^2/2}$ for $0 \le x \lt \pi/2$. [duplicate]

Show that $\cos(x) \le e^{-x^2/2}$ for $0 \le x \lt \pi/2$. This inequality came up in my solution to Show that the sequence $\sum\limits_{k=1}^n\cos\left(\frac kn\right)^{2n^2/k}$ converges. This ...
3
votes
6answers
329 views

limit of $\left( 1-\frac{1}{n}\right)^{n}$

limit of $$\left( 1-\frac{1}{n}\right)^{n}$$ is said to be $\frac{1}{e}$ but how do we actually prove it? I'm trying to use squeeze theorem $$\frac{1}{e}=\lim\limits_{n\to \infty}\left(1-\frac{1}{n+...
3
votes
1answer
114 views

How to prove this series problem: $\sum_{r=1}^\infty \frac{(r+1)^2}{r!}=5e-1$? [closed]

Can I have help with this problem? $$\sum_{r=1}^\infty \frac{(r+1)^2}{r!}=5e-1.$$
3
votes
4answers
102 views

Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$.

Trying to make sense out of the idea that $100\%$ continuous decay is $\frac{1}{e}$, I thought about this: You can express $1+\frac{1}{x}$ as $\frac{x+1}{x}$, such that $\big(1+\frac{1}{x}\big)^x = \...
2
votes
2answers
157 views

Show that $e^{-\beta} = \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \frac{e^{-u}}{\sqrt{u}} e^{-\beta^2 / 4u} du$.

Show that $e^{-\beta} = \frac{1}{\sqrt{\pi}} \int_{0}^{\infty} \frac{e^{-u}}{\sqrt{u}} e^{-\beta^2 / 4u} du$. I'm not really sure of how I should proceed to show this, and it's pretty un-intuitive as ...
2
votes
2answers
264 views

Efficient Method to find funsum$(n) \pmod m$ where funsum$(n)= 0^0 + 1^1 + 2^2 + … n^n$

I have a series: $$f(n) = 0^0 + 1^1 + 2^2 + ..... n^n$$ I want to calculate $f(n) \pmod m$ where $n ≤ 10^9$ and $m ≤10^3$. I have tried the approach of the this accepted answer but the complexity is ...
2
votes
3answers
159 views

Exponential Equation with base -1, 0, and 1

I'm a high school student that learning exponential equations. In my book there is a mathematic problem about exponential equations written as following format : $$h(x)^{f(x)}=h(x)^{g(x)}$$ The ...
2
votes
2answers
1k views

Limit $\lim (1+(a^{1/n}-1) / b)^n$

How can I find the $\lim (1+(a^{1/n}-1) / b)^n$ when $n \to \infty$? I would greatly appreciate it if you kindly give me some hints.
1
vote
2answers
107 views

Why is $a=e$ the smallest number such that $a^x\ge 1+x$ for all $x$? [duplicate]

Calculus book: Find all numbers $a$ such that $\forall x, a^x \ge 1+x$ I immediately thought of the inequality $e^x\ge 1+x$ and guessed that the answer was any number $a$ in $[e,\infty)$. After ...
1
vote
5answers
233 views

Evaluating $\lim_{n \to \infty} n\left(1-\frac{1}{e}\left(1+\frac{1}{n}\right)^{n} \right)$

$$\lim_{n \to \infty} n\bigg(1-\dfrac{1}{e}\bigg(1+\dfrac{1}{n}\bigg)^{n} \bigg)$$ If I write expansion of $\bigg(1+\dfrac{1}{n}\bigg)^{n}$ it was equal to expansion of $e$ so $n-n=0$. Is limit is ...
0
votes
1answer
550 views

Proof that the series expansion for exp(1) is a Cauchy sequence

Consider the series expansion for the exponential function at x = 1: $$a_n := \sum\limits_{i = 0}^n{\frac{1}{i!}}$$ I want to prove that this is a Cauchy sequence, using the remainder formula for ...
0
votes
3answers
247 views

Find the average value of the function $f(t) = 10(e^{5t} - 1)$ over the interval $[0, 1]$.

I have the following solution for my question but I am not sure if I have done the steps right:
0
votes
1answer
139 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 $e = \frac{e^x - e^y}{x-y}$.
-1
votes
1answer
75 views

Prove the Inequality on sequence [closed]

$a_n=(1+\frac{1}{n})^n$ , $b_n=\sum_{k=0}^n \frac{1}{k!}$. Show that $b_n-\frac{3}{2n} < a_n < b_n$.