0
votes
3answers
90 views

Continuity proof for exponential

Show that $f(x) = e^x$ is continuous using the epsilon-delta definition. I can't seem to write down anything meaningful...
1
vote
1answer
23 views

Problems with another characterization of exponential functions

As in other two of my questions, which are already answered by myself, I am treating exponential function again. Now, from the perspective of continuity only. These means, I can not use any single ...
1
vote
2answers
89 views

Show that $\sum\limits_{n=1}^\infty \frac{1}{n^z}$ converges absolutely

I want to show that $\large\sum\limits_{n=1}^\infty \frac{1}{n^z}$ converges absolutely for $\Re(z) > 1$, so I want to show that $\large\sum\limits_{n=1}^\infty |\frac{1}{n^z}|$ converges, or ...
7
votes
2answers
261 views

Proving that $e$ is irrational

Show that $e$ is irrational. Recall $\mathrm{e} = \exp(1)$ so assume $\mathrm{e}$ is rational , then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = \frac{a}{b}\quad \text{for some}\,\,\, a,b \in ...
0
votes
1answer
30 views

Proving properties about complex exponential

I defined $a^z$ for $z \in \mathbb{C}$ as $a^z = \exp(z\log(a))$ and I proved it is continous, now I want to show that $a^n = a \cdot a \cdot a \cdot \ldots \cdot a$ for $n \in \mathbb{N}$ so ...
1
vote
1answer
41 views

Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
0
votes
1answer
53 views

Prove logarithm rules using definition as the inverse exponential

Problem $3$. Show that $\operatorname{exp} : \Bbb R \to (0, \infty)$ is bijective. Its inverse function is called the (natural) logarithm $\log : (0, \infty) \to \Bbb R$. Verify the logarithm ...
0
votes
1answer
65 views

Proving that the exponential function is bijective

Prove that $\exp: \mathbb{R} \mapsto (0,\infty)$ is a bijection. Okay, so the first part is really easy: injectivity follows directly from writing the exponential function as a series. ...
0
votes
3answers
101 views

$0^0$ — indeterminate, or $1$? [duplicate]

One of my teachers argued today that 0^0 = 1. However, WolframAlpha, intuition(?) and various other sources say otherwise... 0^0 doesn't really "mean" anything.. can anyone clear this up with some ...
0
votes
1answer
27 views

Multi part problem to prove functional relation of the exponential function

I'm worried about part (i) right now mostly. Part 3 is easy, and part 2 I can probably get after some work. I know that $\exp(-z) = \large\sum\limits_{n=0}^\infty \frac{-z^n}{n!} = ...
3
votes
1answer
98 views

How can we describe the graph of $x^x$ for negative values?

We usually only see the graph $y=x^x$ for $x>0$, because $x^x$ is a complex number for most negative values of $x$. Yet here is a full graph of $y=x^x$ on the real line: This graph may seem like ...
0
votes
2answers
45 views

$\sup\{a^{r}\mid r<x; r\in\mathbb{Q}\}=\inf\{a^{s}\mid x<s; s\in\mathbb{Q}\}$ How to prove it?

This proposition is a lemma related to another stage for defining exponential function $a^{x}$, in this case for reals, taking into account it is defined for rationals. Proposition Let $a>1$ and ...
0
votes
2answers
32 views

Prove that $(1-\frac{1}{k})^d \le e^{-\frac{d}{k}} $

Prove that $(1-\frac{1}{k})^d \le e^{-\frac{d}{k}} $ for $d,k \ge 0$ I know that $(1+\frac{1}{n})^n \le e$ but does that help? Actually, I don't really 'know' this, but I've heard it's true at least ...
1
vote
6answers
73 views

Noncircular construction of $e$ and $\ln$ for the real line

Could anyone direct me to (or possibly detail) a construction of $e$ and $\ln$ along the reals? For example, they can define $e=\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$ but from this definition ...
2
votes
0answers
37 views

Exponential equations solving methods?

Do you have an idea or general method to solve the following equation?: $$a^{\alpha x}+b^{\beta x} = c^{\gamma x}+ d^{\delta x}$$ when $a,b,c,d$ aren't zero, and $\alpha, \beta, \gamma, \delta$ are ...
2
votes
3answers
134 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
62 views

Show that $\sum_{n=0}^\infty \frac{x^n}{(n!)^2}$ is strictly increasing.

The assignment is: Let $$f: \mathbb{R} \rightarrow \mathbb{R}, x \rightarrow \sum_{n=0}^\infty \frac{x^n}{(n!)^2}$$ and show that, $f\mid_{[0,\infty)}$ is strictly increasing with ...
0
votes
1answer
43 views

Derivation of this inequality [duplicate]

Graphically, it is easy to see but to analytically derive the following inequality, $$|e^x-e^y| \le |x-y| \ \ \ \ \forall x,y \in (-\infty,0] \ \ \ \ (1)$$ do I just apply the MVT and show that ...
2
votes
1answer
105 views

Different proofs of $\lim_{x\to \infty}\left(1+ \frac{1}{n}\right)^n =e$

I recently was teaching my friend about the number $e$. I introduced him the number by using the compound interest thing . Then I wrote down the general result -$$\lim_{x\to \infty}\left(1+ ...
4
votes
3answers
102 views

Conclusion about limit definition of e^a for a sequence of real numbers {a_n} converging to a?

I have seen this fact used in several demonstrations, but have never seen a proof of it. I believe the statement is: If $\{a_n\}$ is a sequence of real numbers such that $a_n \rightarrow a$ finite, ...
3
votes
2answers
71 views

Inverse of the derivative for f(x) = f'(x)

I'm new so forgive my inexperience here. The problem concerns the following: $$ f: \Bbb R \to (0, \infty), f(x) = f'(x) $$ The first part of the problem involves showing f is increasing, this ...
3
votes
2answers
508 views

When is the sum of two exponentials functions equal to another exponential function?

Fix real numbers $a_1$, $a_2$, $a_3$ and $b_1$, $b_2$, $b_3$ and $c_1$, $c_2$, $c_3$. Consider the equation $$ a_1\exp(b_1 (x-c_1)) + a_2\exp(b_2 (x-c_2)) = a_3\exp(b_3 (x-c_3)) $$ in $x$. My ...
2
votes
0answers
194 views

How to prove $x^y$ is jointly continuous?

It's known that real exponentiation $x^y$ is continuous in each variable, but is real exponentiation jointly continuous in both the exponent and the base? I considering the function ...
3
votes
3answers
333 views

Exponential function and uniform convergence of polynomials.

How can I prove that no sequence of polynomials converges uniformly to the exponential function? Thanks in advance for any help.
9
votes
2answers
224 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 ...
5
votes
4answers
79 views

Multiplying the long polynomials for $e^x$ and $e^y$ does not give me the long polynomial for $e^{x+y}$

As an alternative to normal rules for powers giving $e^xe^y=e^{(x+y)}$ I am multiplying the long polynomial of the taylor series of $e^x$ and $e^y$. I only take the first three terms: $$ ...
3
votes
1answer
94 views

How to show limit definition of $e^{z}$ holds if $z \in \mathbb{C}$

It is well known that for $x\in \mathbb{R}$ we have $$ e^{x} = \lim_{n \to \infty} \left(1+ \frac{x}{n}\right)^n. $$ This follows quickly by considering logarithms and using L'Hospital's rule. ...
2
votes
2answers
99 views

A non-zero function satisfying $g(x+y)=g(x)g(y)$ must be exponential function

Let $g$ be a non-zero function satisfying $g(x+y)=g(x)g(y)$. Show that the function must be exponential function.
19
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 ...
2
votes
2answers
171 views

How to find $f(x)+f(x+1) = e^x$?

Let $x,a,b$ be real numbers and $f(x)$ a (nongiven) real-analytic function. How to find $f(x)$ such that for all $x$ we have $f(x)+af(x+1)=b^x$ ? In particular I wonder most about the case $a=1$ ...
4
votes
2answers
1k 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
3answers
88 views

Minimizing bizarre exponential function

I am trying to minimize the function in the form of $f(x) = (1-a^x)^x$ where $0 < a < 1$ with respect to $x$ (for $x > 0$) and I am stuck! Unfortunately the derivative is not nice enough to ...
4
votes
2answers
142 views

Solve equation $\tfrac 1x (e^x-1) = \alpha$

I have the equation $\tfrac 1x (e^x-1) = \alpha$ for an positive $\alpha \in \mathbb{R}^+$ which I want to solve for $x\in \mathbb R$ (most of all I am interested in the solution $x > 0$ for ...
2
votes
1answer
147 views

A functional equation related to the exponential function

Suppose that $f:\mathbb{R}\rightarrow (0,\infty)$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ are two functions and satisfies the following relation \begin{equation} f(xy)=f(y)^{g(x)} \end{equation} ...
2
votes
3answers
113 views

Find an exponential function with given condition

How can I have an example of an exponential function defined in the X range 1 - infinity, with values starting at 40 and converging to 1?