Tagged Questions

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Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt$$ This could be used to evaluate partial sums using knowledge of the ...
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Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
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Maclaurin polynomial for $\arcsin(x)$

How would I find the 3rd-order Maclaurin polynomial for $f(x) = \arcsin(x)$; with the interval $(0,\frac 3 4)$ to show it in terms of $x$? Would you have to somehow manipulate it to $\dfrac{1}{1+x}$ ...
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Find all $\alpha \in \mathbb{R}$ such that $\int_0^\infty \sin(x^\alpha)dx$ converges.

Find all $\alpha \in \mathbb{R}$ such that $\int_0^\infty \sin(x^\alpha)dx$ converges. There is an answer here that differs from mine (they claim for $-\infty<\alpha<-2$ and ...
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Taylor series in order to find the approximate antiderivative of a function

Somewhat inspired by this question about antiderivatives, I started to check whether or not that function had an elementary antiderivative. Then, after checking with Maxima, it struck me that, by ...
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Maclaurin Series of $\int_0^x \cos t^2\,dt$

Find the Maclaurin Series for $\int_{0}^{x}\cos t^2\,dt$. $$\cos(x) = \sum\frac{(-1)^n x^{2n}}{2n!}$$ I'm trying this: $$\cos^2 x = \sum\frac{(-1)^n x^{4n}}{(2n!)^2}$$ How would you solve this ...
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Evaluating $\int_{0}^{\frac{\pi}{2}} \arctan( a \sin x) \ dx$ using the Taylor expansion of $\arctan (x)$

I was wondering if it's possible to show that for $a >0$, \begin{align}\int_{0}^{\pi/ 2} \arctan (a \sin x) dx &= 2 \sum_{k=0}^{\infty} \frac{\left(\frac{\,\sqrt{\vphantom{\Large A}\,1 + ...
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Series around $s=1$ for an integral
Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...
Approximate the integral $\int_0^1 \sin(x^2) dx$
I'd like to ask if someone can please give me a little push with this assignment: Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations ...