This might be an extremely stupid question but I stuck on two different definition of series expansion of $\ln(1+x)$. In this Resonance article, the author assert that - $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots=\ln 2$$ based on the series expansion $\ln(1+x)=x-\frac{x^2}{2}+\fr...

I'm trying to calculate the derivative of $f(x) = \|x\|_{g(x)}^2$, where $g$ is a Riemannian metric on $\mathbb{R}^n$. I calculate the derivative of direction $h$: $$\begin{align}\lim_{s\to 0} \frac{f(x+sh)-f(x)}{s} &= \lim_{s\to 0} \frac{\langle x+sh, x+sh \rangle_{g(x+sh)} - \langle x, x\rangle...

I'm given a set of $16$ points $(d,t)$ where $d\in\{0,3,10,30\}$ and $t\in\{2,5,10,15\}$. $r$ is a function of both $d$ and $t$. $9$ measurements of $r$ are taken at each $(d,t)$ point. How can I then find the standard error associated with the trapezoidal rule for $\int r \,\mathrm{d}t $? I've ...

I'm learning how to derive standard error of the mean mathematically. I understand that standard error is the square root of the variance of the mean: $SE =\sqrt {Var((x_1+x_2+...+x_n)/n}$ And understand $$SE = 1/n \sqrt {(Var((x_1+x_2+...+x_n))}$$ $$SE = 1/n \sqrt {(Var(x_1)+Var(x_2)+...+Var...