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

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10
votes
5answers
757 views

Solve the equation $2^x=1-x$

Solve the equation: $$2^x=1-x$$ I know this is extremely easy and I know the solution using graphical approach. Basically, I can see the solution, but I can't work it out algebraically.
10
votes
2answers
485 views

Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series

In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following: $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over ...
12
votes
3answers
449 views

Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?

In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one? I can of course ...
7
votes
7answers
1k views

About $\lim \left(1+\frac {x}{n}\right)^n$

I was wondering if it is possible to get a link to a rigorous proof that $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
6
votes
3answers
641 views

Finding asymptotes of exponential function and one-sided limit

Find the asymptotes of $$ \lim_{x \to \infty}x\cdot\exp\left(\dfrac{2}{x}\right)+1. $$ How is it done?
21
votes
13answers
1k 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 ...
31
votes
18answers
3k views

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
3
votes
1answer
190 views

How can one prove the impossibility of writing $ \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions?

Can we express $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions? (Note: Infinite series are not allowed.) If not, then is there a proof that $ \displaystyle \int ...
3
votes
1answer
158 views

Can the graph of $x^x$ have a real-valued plot below zero?

The function $f(x) = x^x$ gives a complex number only if x has an even denominator. I'm not sure about irrational numbers. Why, then, is the best graph I can find of that function that of Wolfram ...
17
votes
3answers
224 views

What combinatorial quantity the tetration of two natural numbers represents?

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. ...
21
votes
9answers
1k views

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 ...
7
votes
3answers
387 views

How was $e$ first calculated?

I understand how $\pi$ is calculated, but I am interested in references that explain when and how the natural exponent $e$ was developed. What mathematical principles are behind the value of $e$?
20
votes
18answers
2k views

How to understand why $x^0 = 1$, where $x$ is any real number?

Alright, so the idea of an exponent, $x$, is that you are multiplying its base by itself $x$ number of times. With base $5$ and $x=3$, we have that $5^3$ = $5 \cdot 5 \cdot 5$ I understand that the ...
6
votes
3answers
222 views

Prove that $e^{-A} = (e^{A})^{-1}$

Let $A, B \in R^{n \times n}$. Prove that $e^{-A} = (e^{A})^{-1}$. ($R$ is the real numbers) I've tried messing around with both sides, evaluated as sums. I just can't get the two to match up. Any ...
3
votes
3answers
208 views

Is it trivial to say $\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k}$

Is it trivial to say $$\mathop {\lim }\limits_{n \to \infty } {(1 + {k \over n})^n} = e^{k},$$ considering the fact that we know $$\mathop {\lim }\limits_{n \to \infty } {(1 + {1 \over n})^n} = e?$$ ...
38
votes
4answers
1k views

If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?

It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true? If $e^A$ and $e^B$ commute, do $A$ and $B$ commute? Edit: Addionally, what ...
12
votes
4answers
344 views

Euler's identity: why is the $e$ in $e^{ix}$? What if it were some other constant like $2^{ix}$?

$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis. My intuition behind this is that ...
10
votes
2answers
1k views

Show that $ e^{A+B}=e^A e^B$

If $A$ and $B$ are $n\times n$ matrices such that $AB = BA$ (that is, $A$ and $B$ commute), show that $$ e^{A+B}=e^A e^B$$ Note that $A$ and $B$ do NOT have to be diagonalizable.
4
votes
2answers
333 views

Closed form for $n$-th derivative of exponential

I need the closed-form for the $n$-th derivative ($n\geq0 $): $$\frac{\partial^n}{\partial x^n}\exp\left(-\frac{\pi^2a^2}{x}\right)$$ Thanks! By following the suggestion of Hermite polynomials: ...
13
votes
6answers
425 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
7answers
351 views

How to prove continuity of $e^x$.

I simply want a proof that $e^x$ is continuous. I have never really been able to find something satisfying these points: $e$ is defined to be the limit $\lim_{n\to\infty}\left(1+{1\over ...
6
votes
9answers
3k views

Proving $\lim \limits_{n\to +\infty } \left(1+\frac{x}{n}\right)^n=\text{e}^x$.

I knew that $e^x=\lim \limits_{n\to+\infty }{\left(1+\frac{x}{n}\right)^n}$. But I've never seen its proof. So I tried to prove it using $\exp(\ln x)=\ln(\exp(x))=x$. Here is what I've tried so far : ...
5
votes
3answers
103 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: $$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$ $$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$ $$ = ...
3
votes
1answer
175 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 ...
2
votes
5answers
658 views

How does $e^{i x}$ produce rotation around the imaginary unit circle?

Euler' formula states that: $e^{i x} = \cos(x) + i \sin(x)$ I can see from the MacLaurin Expansion that this is indeed true, however, I don't intuitively understand how raising $e^{i x}$ power ...
6
votes
7answers
1k views

How would you explain why “e” is important? (And when it applies?) [duplicate]

Possible Duplicate: Intuitive Understanding of the constant “e” Let's say you want to explain this to your teenage son. I understand the technical definition of $e$ $$ ...
4
votes
3answers
234 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 ...
3
votes
3answers
700 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 ...
2
votes
1answer
60 views

The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

This question come from a recent post Exponential extension of $\mathbb{Q}$. An exponential field is a field $\mathbb{K}$ where it's well defined a function $E:\mathbb{K} \rightarrow \mathbb{K}$ ...
1
vote
3answers
199 views

How to solve the definite integral $\int_{-4}^{-2}e^{-x}\,dx$?

I'm trying to find the value of the integral $\int_{-4}^{-2}e^{-x}\,dx$ but I just couldn't solve it. Actually I found in a List of integrals that $\int e^x\,dx=e^x+C$ so I concluded: ...
1
vote
2answers
341 views

Power Series Expansion Problem Analysis

Please show that for all $x,y\in\mathbb{R}$, $$e^{x+y} - e^xe^y = \lim_{k\to\infty} \sum_{n=1}^k \sum_{j = 0}^n\left(\frac{x^{k+j}}{(k+j)!}\frac{y^{n-j}}{(n-j)!} + ...
10
votes
2answers
213 views

Conjectural closed form for $\int_0^\infty\sqrt[3]z\ \operatorname{Ei}^2(-z)\,dz$

While trying to answer the question "A closed form for $\displaystyle\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$", I came up with a conjecture: $$\int_0^\infty\sqrt[3]z\ ...
13
votes
4answers
654 views

Prove that e is irrational

Prove that e is an irrational number. Recall that e $=\displaystyle\sum_{n=0}^\infty\frac{1}{n!},\,\,$ and assume $\mathrm{e}$ is rational. Then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = ...
17
votes
10answers
1k views

Why isn't $\lim \limits_{x\to\infty}(1+\frac{1}{x})^{x}= 1$?

Given $\lim \limits_{x\to\infty}(1+\frac{1}{x})^{x}$, why can't you reduce it to $\lim \limits_{x\to\infty}(1+0)^{x}$, making the result "$1$"? Obviously, it's wrong, as the true value is $e$. Is it ...
9
votes
2answers
172 views

Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$

This comes from the comments section of this question here, credits Lucian. The statement is $$\sum_{n=1}^\infty\left(e-\sum_{k=0}^n\frac1{k!}\right)=1$$ This looks really interesting, so I was ...
10
votes
1answer
190 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
7
votes
2answers
108 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
4
votes
2answers
2k views

Why is $e^{x}$ not uniformly continuous on $\mathbb{R}$?

It seems intuitively very clear that $e^{x}$ is not uniformly continuous on $\mathbb{R}$. I'm looking to 'prove' it using $\epsilon$-$\delta$ analysis though. I reason as follows: Suppose $\epsilon ...
2
votes
1answer
508 views

Commuting in Matrix Exponential

Let $A$, and $B$ be commuting $n\times n$ matrices, i.e. $A.B = B.A$. Let \begin{equation} \exp(A) = \sum_{i=0}^\infty\frac{1}{i!} A^i \end{equation} show that $\exp(A+B) = \exp(A).\exp(B)$.
6
votes
3answers
135 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 ...
5
votes
5answers
124 views

Finding the derivative of $2^{x}$ from first terms?

I was trying to understand why $e^{x}$ is special by finding the derivatives of other exponential functions and comparing the results. So I tried ${\rm f}\left(x\right) \equiv 2^{x}$, but now I'm ...
3
votes
2answers
60 views

Proof of the derivative of $a^x$ [duplicate]

I've tried for a while myself from first principles and applying various rules, but always end up going in circles. I've gotten as far as $$ y = a^x $$ $$ \frac{dy}{dx} = a^x \left( \lim_{x ...
3
votes
2answers
1k views

Integral of exponential function with polynomial argument

I am looking for an effcient way to evaluate $$ I = \int_{-\infty}^\infty dx\ e^{-ax+bx^2+cx^4}\\ \text{where } a,b,c \in \mathbb{R}^+ $$ I have already read about a solution involving the series ...
2
votes
5answers
254 views

Does the series $\,\displaystyle\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?

I'm trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart's Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} - 1)$$ I've considered ...
2
votes
6answers
377 views

If $a_1,\ldots,a_n>0$ and $a_1+\cdots+a_n<\frac{1}{2}$, then $(1+a_1)\cdots(1+a_n)<2$.

Assume that $a_1,a_2,\ldots,a_n>0$ and $a_1+a_2+\cdots+a_n<\frac{1}{2}$, and prove that $$(1+a_1)(1+a_2)\cdots(1+a_n)<2$$ I've tried Hölder's inequality (the same result can easily be ...
2
votes
3answers
346 views

Prove that the limit definition of the exponential function implies its infinite series definition.

Here's the problem: Let $x$ be any real number. Show that $$ \lim_{m \to \infty} \left( 1 + \frac{x}{m} \right)^m = \sum_{n=0}^ \infty \frac{x^n}{n!} $$ I'm sure there are many ways of pulling this ...
1
vote
2answers
130 views

Exponential function, multivariable calculus

I got the function $f(x,y)=e^{-x^2-y^2}$ with the domain of definition $x^2+y^2\leq25$. The task is to decide the biggest and lowest value. How do I get there?
1
vote
1answer
94 views

Since $2^n = O(2^{n-1})$, does the transitivity of $O$ imply $2^n=O(1)$?

Let us assume that $f(n)=2^{n+1}$, $g(n)=2^n$ be two functions. Now, use limit to find $O(f(n))$: $\lim_{n\to\infty} \dfrac{2^{n+1}}{2^n}=2$. This is not equal to infinity, so the limit exists, hence ...
1
vote
3answers
137 views

Definition of hyperbolic cosine and its relation with exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula ...
1
vote
1answer
98 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 ...