0
votes
1answer
31 views

Arp Equation, simple, b = 0

I am working with Arp Equations. Letters slightly changed for simplicity/formatting purposes. \begin{equation} q(t) = \frac{A}{(1+bDt)^{1/b}} \end{equation} now when $b = 0$, how does the result end ...
5
votes
2answers
88 views

Integral $\int_0^{\infty} \log(x) e^{-x^2} \mathrm{d}x = -\frac{1}{4}\sqrt{\pi} (\gamma + \log(4)).$

While trying to compute the expected value $E[\log(X)]$ for a normally distributed variable $X$ I found the following integral $$\int_0^{\infty} \log(x) e^{-x^2} \mathrm{d}x = -\frac{1}{4}\sqrt{\pi} ...
1
vote
4answers
413 views

Explain this chain rule for differentiating $y=xe^{-kx}$

I am asked to differentiate $$y=xe^{-kx}$$ The answer I am given is $$e^{-kx}(-kx+1)$$ I understand that when e is differentiated, it remains the same. I also see the product rule, but I'm not sure ...
8
votes
1answer
235 views

Is this Euler-Mascheroni constant calculation from double integrals a true identity?

A prime number is a number that is only divisible by itself and one, that is the number of divisors of a prime number is equal to $2$. One way to illustrate this is to plot a matrix such that if the ...
2
votes
3answers
109 views

Limit of an equation similar to the Euler's constant definition

$$ \lim_{n\to \infty} \left(2+\frac{1}{n}\right)^{n} = ? $$ I don't know even how to start this.
1
vote
1answer
131 views

Use Euler's method with step size 10^-n to estimate x(1), where f(x) is the solution of the initial-value problem below. f(x)=-x x(0)=1

Use Euler's method with step size $10^{-n}$ for $n=1,2,3,4.$ to estimate $x(1)$, where $f(x)$ is the solution of the initial-value problem below. $x'=f(x)=-x$ $x(0)=1$ EDIT / UPDATE: x_n+1=x_n + ...
3
votes
2answers
160 views

Potence of Euler's Number

Show with help of the Bernoulli Inequality that $$\lim_{n\rightarrow\infty}\left(1-\frac{1}{n^2}\right)^{n}=1$$ End with: $$\lim_{n\rightarrow\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}$$
4
votes
4answers
368 views

Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?

Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication? I had also got a clue: it's related to e. Please help! ...
3
votes
2answers
195 views

Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant

Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution ) ...
5
votes
2answers
408 views

Equality with Euler–Mascheroni constant

While trying to prove integral with exponential function and logarithm in an alternative way, I came to this solution: $$\sum_{k=0}^{+\infty}(-1)^{k+1}\frac{\log (k+1)+\gamma }{(k+1)}.$$ As both ...