# Tagged Questions

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

28 views

69 views

39 views

### Exponential matrix decay

I’m working on contractive systems that have a system of ODE equations. I have an exponential matrix multiply by time that is for a given matrix $A$ I’m getting $e^{At}$. I want to know what are the ...
76 views

89 views

### What is the evaluation of $\sum_{k=1}^{\infty}\frac{k^n}{k!}$? [duplicate]

I stumbled upon a similar problem and really liked the answers there, so I wondered if there were a general solution for $$\sum_{k=1}^{\infty}\frac{k^n}{k!}=?$$ Sadly, when I try to apply some of ...
34 views

### What function describes this problem of every possible breeding of a set of dogs?

If I have n dogs [a, b, c, ...], and I want to breed them in every possible combination (every possible binary tree made of ...
109 views

### An integral for $2\pi+e-9$

Motivation Lucian asked about the almost-integer $2\pi+e\approx9$ in a comment to a partially answered why question about $e\approx H_8$. This is more involved than approximations to $\pi$ and ...
119 views

### Show $\sum_{k=1}^\infty \frac{k^2}{k!} = 2\mathrm{e}$ [duplicate]

I stumpled upon the equation $$\sum_{k=1}^\infty \frac{k^2}{k!} = 2\mathrm{e}$$ and was just curious how to deduce the right hand side of the eqution - which identities could be of use here? Trying ...
54 views

### What are the solutions for $2^x=x^2$? [duplicate]

What are the solutions for $2^x=x^2$? I noticed there were 2 roots: $2,4$. Are there any other roots, and how do you calculate them?
30 views

### Solving an expression containing two added exponential functions

I have a problem solving the below equation with respect to $x$: $0.6\cdot \exp(\frac{-40}{x})+0.4 \cdot \exp(\frac{10}{x})=1$ My problem is that I have two exponential functions which are added ...
102 views

### Limit of $(1+3/n)^{4n}$ as $n$ goes to infinity

This afternoon I was trying to evaluate $$\lim\limits_{n \to \infty} \biggl( 1 + \frac{3}{n}\biggr)^{4n}$$ but was having some difficulty in doing so. I know the answer to be $e^{12}$, and can ...
49 views

### Simplify $\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$

How do you make the jump from: $$\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$$ To: $$\frac{25^{21}-4^{21}}{25^{21}-4(25^{20})}$$
89 views

### Prove Exponential Function Inequality: $e^x \le \frac{1}{1-x}$

Prove that $e^x \le \dfrac{1}{1-x}, x\lt 1.$ I find that if we set $f(x)=e^x(1-x)$ then $f(0)=1$ and $f'(x)<0, x\in(0,1]$ proving the inequality for $x\in[0,1]$ but I don't see how to prove it ...
50 views

### how tell if a series of power numbers is bigger then others

I trying to order a list of mathematical expressions in string format as: "2*2" "4^1" "4^2^5" so far, so good for non exponential operations (^). I could compute ...
67 views

### How can one properly understand the fact that $e^x$ can be differentiated an infinite amount of times?

Simply put if I follow the rule derived by the simple proof denoting $e^x$ to be the derivative of $e^x$ then it follows that it should have an infinite number of derivatives. Is this a conceptual ...
487 views

### Number system with $e^x = 0$ for some $x$

It is well known that $e^x \ne 0$ for all $x \in \mathbb{R}$ as well as $x \in \mathbb{C}$. Upon reading this article and doing a bit of research I have found that this also applies to the ...
32 views

### Limit of a sequence of functions recursively defined by integrals

$f_n:[0,\infty)\to\mathbb{R}$ is defined recursively by $f_1:=0$ and $$f_{n+1}(x)=e^{-2x}+\int_0^xf_n(t)e^{-2t}dt,\qquad n\ge 1$$ I need to show that the limit $f(x):=\lim_{n\to\infty} f_n(x)$ exists ...
79 views

### Square root of $e^{ix}$

Is this $$\sqrt{e^{ix}}=e^{\frac{ix}{2}}=\cos\frac{x}{2}+i\sin\frac{x}{2}$$ true? Or $$\sqrt{e^{ix}}=\sqrt{\cos x+i\sin x}$$ How do I express square root of $e^{ix}$ as a non-square root expression?
81 views

I want to show that $$\lim_{n \to \infty} \left(1-\frac{n}{n^2} \right) \left(1-\frac{n}{n^2-1} \right) \cdot \ldots \cdot \left(1-\frac{n}{n^2-n+1} \right) = \lim_{n \to \infty} \prod_{k=0}^{n-1} \... 2answers 60 views ### Proof that \lim_{x\to\infty} b^x=0 \iff 0 \leq b<1 Are there any errors in the following attempt to prove the above? (\Leftarrow) Let f(x)=b^x, with 0 \leq b<1. Then, for all x, f(x)>0 and f'(x)=b^x \ln(b)<0. This means that f ... 4answers 81 views ### Complex Finite Product \prod_{k=0}^{n-1} (1-\zeta^k z) I am working on a review for a graduate level Complex Analysis course. The following problem is on the review: Let \zeta= e^{\frac{2\pi i}{n}} (n\in \mathbb{N}); show that \displaystyle{\prod_{... 2answers 63 views ### Show that \exp(-\lambda x) \cdot\exp(\lambda x)=1 using the power series Let A be a commutative Banach algebra. Consider the exponential function$$\exp(\lambda x) = \sum_{n=1}^\infty\frac{(\lambda x)^n}{n!}, where $x \in A$ and $\lambda \in \mathbb C$. We can easily ...
How do you solve $10^x = x$? I'm not sure how to solve this algebraically. Using log functions wasn't enough.
### Provided that $f(u)$ is holomorphic, prove that $u$ is constant
Let $u: \mathbb{C} \rightarrow \mathbb {R}$ be a real valued function, so that $f(z)=\cos \left( u(z) \right) +i \cdot \sin \left( u(z) \right)$ is holomorphic in $\mathbb{C}$. Show that $u$ is ...