How to derive this series February 14    1

I am wondering how to derive this series?I know that But not sure how to solve the first one. Please give me some hints. Thanks a lot.A.EDIT:$$\sum_{r=1}^\infty r x^{r-1}=\sum_{r=1}^\infty \frac{d (x^r)}{dx}=\frac{d\left(\sum_{r=1}^\infty x^r\right)}

Calculating a growing series in Spreadsheet February 14    1

I've got a spreadsheet, where I'm trying to calculate the amount of retained users over time for a subscription based service. can get the totals but it's

What does the notation $B(x;\epsilon)$ mean February 14

What does the notation $|B(x,\epsilon)|$ mean? Is it the volume of the open ball? I saw it from the following:...It follows from the Lebesgue theory that for almost every $x\in\Omega$, $\lim_{\epsilon\to 0} \frac{1}{|B(x,\epsilon)|} \int_{B(x,\epsilo

Inverse Gaussian integration February 14

I tried to derive the option price under a inverse Gaussian distribution, in which I encounter the following integral and don't know how to solve $$\int_0^kx f(x;\lambda,\mu)dx=\int_0^kx \Big(\frac{\lambda}{2\pi x^3}\Big)^{1/2}exp\frac{-\lambda(x-\mu

Cartesian Product converted into Summation February 14

I am looking at the proof of Maximum Likelihood Estimator and So let's get to it: first take the $\log$ of the equation: $$\log(P(\text{DATA}))=\log\prod i=1N(PX_i(1−P)1−X_i)$$Since $$\log(a b)=\log(a)+\log(b)$$then all the terms of the product becom

Inverse Gaussian, Limiting Distributions February 14    1

I'm trying to understand the nature of limiting distributions and distributions, specifically$1/Z_n \longrightarrow ~?$ where $Z_n\longrightarrow Z -Gaussian(0,1)$I understand that the gamma distribution converges to the gaussian for a large enough $

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