Tagged Questions

Questions regarding the Taylor series expansion of univariate and multivariate functions, including coefficients and bounds on remainders. A special case is also known as the Maclaurin series.

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zero of second order

I'm studying functions associated with a domain in the complex plane. In one paper that I'm reading, a particular function, $R(a, b)$, is discussed (with "$a$" varying and "$b$" fixed complex ...
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Intuition for polynomial bases

In my linear algebra course I stumbled upon the following observations. We have some function $f: \Bbb{R} \to \Bbb{R}$, $f = f(x)$. $f(x)$ may be composed of elementary functions or not, but in ...
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Geometric proof of expansions of series

I have read that Barrow had proved the fundamental theorem of calculus. I have read that proof and its a good. Further I know Newton had derived the sine and cosine series. His methods obviously didn'...
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Taylor series of complex function

$f(z) = \frac{2}{z^2-1}$ at $z = i$ My solution: $t = z - i$ $z = t + i$ $\frac{2}{z^2-1} = -2\frac{1}{1-(t+i)^2} = -2\sum_{n=0}^\infty (t+i)^{2n} = -2\sum_{n=0}^\infty z^{2n}$ Where is my ...
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Perturbation: compute an approximation to the solution of the equation $y+\epsilon\sin y=x^2$

Compute approximation to the solution of the equation $y+\epsilon \sin y=x^2$ using perturbation method. Assume that terms involving powers of $\epsilon$ of order 3 or more can be ignored. So far I ...