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bio website reddit.com/r/…
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visits member for 2 years, 4 months
seen 6 hours ago

when someone smiles at me, all I see is an ape bearing its teethe


Jul
16
comment Show that $(x+1+O(x^{-1}))^x = ex^x + O(x^{x-1})$ for $x\rightarrow \infty$
So when we say $g=O(f)$ what we're saying is g is some element of the set of all functions whose absolute value is bounded by a constant multiple of $f$, is that a correct understanding?
Jul
16
comment Show that $(x+1+O(x^{-1}))^x = ex^x + O(x^{x-1})$ for $x\rightarrow \infty$
why can't you then just set A to be the max of your bounded function?
Jul
16
comment Show that $(x+1+O(x^{-1}))^x = ex^x + O(x^{x-1})$ for $x\rightarrow \infty$
Also, can I essentially just replace $O(f(x))$ with $Af(x)$ for $A$ some unknown positive constant?
Jul
16
comment Show that $(x+1+O(x^{-1}))^x = ex^x + O(x^{x-1})$ for $x\rightarrow \infty$
Thank you Norbert. Could you explain your justification for getting rid of the log in the third equals sign?
Jul
16
asked Show that $(x+1+O(x^{-1}))^x = ex^x + O(x^{x-1})$ for $x\rightarrow \infty$
Jul
16
comment Help Proving that $\frac{(1+\frac{1}{t})^t}{e} = 1 -\frac{1}{2t} + O(\frac{1}{t^2})$ for $t\geq 1$
@did: could you explain the missing step for me?
Jul
16
accepted Help Proving that $\frac{(1+\frac{1}{t})^t}{e} = 1 -\frac{1}{2t} + O(\frac{1}{t^2})$ for $t\geq 1$
Jul
16
comment Help Proving that $\frac{(1+\frac{1}{t})^t}{e} = 1 -\frac{1}{2t} + O(\frac{1}{t^2})$ for $t\geq 1$
Thanks for this, very slick, but could you please quickly explain how you derived the final equality?
Jul
16
asked Help Proving that $\frac{(1+\frac{1}{t})^t}{e} = 1 -\frac{1}{2t} + O(\frac{1}{t^2})$ for $t\geq 1$
Jul
14
comment Why doesn't $2\pi\int_{-1}^1\sqrt{1-x^2}dx$ give the surface area of a sphere of radius $1$?
Thanks for the comments and links, they've been a big help.
Jul
14
asked Why doesn't $2\pi\int_{-1}^1\sqrt{1-x^2}dx$ give the surface area of a sphere of radius $1$?
Jul
12
revised Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
deleted 77 characters in body
Jul
12
comment Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
You're right, my mistake, apparently I should be more wary of Wolfram's margin of error.
Jul
12
comment Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
I checked numerically that removing the 1 makes the inequality untrue, I don't think you can in general integrate the function $|\sin\left(\frac{\pi}{t}\right) -\frac{\pi}{t}\cos\left(\frac{\pi}{t}\right)|$ without first pulling the absolute value bars outside the integral; consider $|x|$ in an interval around zero.
Jul
12
accepted Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
Jul
12
comment Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
Thank you for this!
Jul
12
comment Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
What it looks like is an elliptic integral (I know nothing about these, beyond apparently what they look like).
Jul
12
comment Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
Do Carmo's "Differential Geometry of Curves and Surfaces" Section 1-3 problem 9b, the integral is the arc length of a portion of the curve $(t,tsin(\frac{\pi}{t}))$.
Jul
12
revised Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
deleted 132 characters in body
Jul
12
comment Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
Does anyone have any ideas? I can't believe this thing is so difficult.