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 Jul23 comment Does the series $\sum\limits_{n=1}^\infty \frac{1}{n\sqrt[n]{n}}$ converge? +1 very simple ! Jul23 comment Simple Logarithms Equation Assuming $x\in\mathbb{R}$, and since $3^x>0$ for all $x\in\mathbb{R}$, then $3-x>0$ which implies $x<3$. Jul22 comment If I buy 2 lottery tickets do I double my chance of winning? Which gives $$\frac{1}{{49\choose 6}}=0.00000007151123842...$$ I'd better get a better paid job. Jul6 comment Plot of $n$ concentric circles at once? Don't you mean $-i^2$ in your third equation, and not $-n^2$ ... ? PS great question ! Jun28 comment Integral with only a list of values I was a bit vague about the meaning of "under" because: suppose $a0$... math.stackexchange.com/questions/256992/…... an improvement on $\Re(s)>1$, but still not the whole story ! Jun24 comment My brother asked me to explain a algebra problem. How should I explain it? You're very welcome ! Welcome to SE. Jun21 comment Why $\sum\limits_{n=1}^\infty \frac{e^{inx}}{n}=-ln(1-e^{ix})$ in $D'$ Not quite sure about the question, but maybe this will help: Your sum is just $\sum_{n=1}^\infty\frac{(e^{ix})^n}{n}$, and $\sum_{n=1}^\infty \frac{z^n}{n}$ is a well known power series... Jun9 comment Why is $e$ so special? Thinking aloud: the function $e^x$ has many interesting properties, e.g. if we consider the function $e^x$, then we find the derivate of $e^x$ is itself $e^x$. This is quite remarkable ! When $x=1$ we obtain $e^1=e$. $e^x$ is also strictly positive for all $x\in\mathbb{R}$. $e^x$ is also a transcendental function. Another remarkable formula involving $e^x$ is $e^{i\pi}+1=0$, where $i=\sqrt{-1}$. Jun8 comment The value of $\int^{\pi/2}_0 \frac{\log(1+x\sin^2\theta)}{\sin^2\theta}d\theta$ Sorry my mistake - updated. Jun2 comment Why do we assume the complex plane is curvey at infinity? @ Ollie Ford yes, I think that's what popped into my head, but mathematics needs to be rigorous, so this is just food for thought really ! Jun2 comment Why do we assume the complex plane is curvey at infinity? Just a thought: Give me any point $z$ in the upper half complex plane and I can tell you that the semi-circular contour of radius $|z|+1$ encloses it. Rectangular regions can be constructed similarly. Jun2 comment Extension of $|\cdot|_\infty$ on $\mathbb R$ to $\mathbb C$ I think this is related to Ostrowski's Theorem. Check out en.wikipedia.org/wiki/Ostrowski%27s_theorem "...any field, complete with respect to an archimedean absolute value, is (algebraically and topologically) isomorphic to either the real numbers or the complex numbers. This is sometimes also referred to as Ostrowski's theorem." Jun1 comment Name for norm with property $\|x+y\|=\|x\|+\|y\|$. Obviously people are liking your answer, but could you please clarify: Are you saying such a norm is called a "trivial norm"? May31 comment How do I calculate this integral:$\int_{0}^{1}\ln^2 \left| \sqrt x-\sqrt{1-x} \right|dx$? I would split the domain to which $x$ belongs to describe positive and negative regions. That way you can dispose of the absolute value and simply evaluate a finite sum of integrals having no absolute values.