# Questions tagged [taylor-expansion]

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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### Show that $\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1$

Show that $$\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1 .$$ I know that $\arctan 1 = \frac{\pi}{4}$ and that the sequence ...
1answer
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### Laurent series 1/(z+2)+1/z^2, 0<|z+2|<2

I can't find a way to represent $z^2$ in terms of $z+2$. I've tried to do $(z+2)(z-2)+4$, but I'm stuck with the $+4$.
0answers
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### Relative position of a function's surface and a tangent plane

Let $f$ be a function such that \begin{align*} f(x,y)=&f(x_0,y_0)+(x-x_0)\frac{\partial f}{\partial x}(x_0,y_0)+(y-y_0)\frac{\partial f}{\partial y}(x_0,y_0)+ \\ &\frac{1}{2}\left[ (x-a)^2\...
0answers
21 views

### Laurent Expansion of a Composition (with the inverse)

Suppose $f(z) = a_1z + a_0 + O(\frac{1}{z})$ as $|z| \rightarrow \infty$. Here $a_1 > 0$. Let $W(z) = z + \frac{1}{z}$. How do I obtain that the composite function $g(z) = W \circ f^{-1}(z)$ has ...
2answers
42 views

### Finding Taylor Series of the following function?

How do I go about finding the answer to this? I have tried two methods, using the known taylor series for $\sin (x)$ and $\frac{1}{1-r}$ and then transform and add those, and I have also tried by ...
0answers
34 views

### Is it possible to compute this infinite series explicitly? [on hold]

Is it possible to compute explicitly or in an approximate way the infinite series: $$S(x)=\sum_{n=0}^{\infty}\frac{a^{n}}{n!}f^{(2n+1)}(x),$$ for a generic function $f$?
4answers
62 views

1answer
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### Minimum value of $n$ for Lagrange reminder on Taylor polynomial of $\frac{1}{x}$

I'm trying to solve the following question: "Find the minimum value of $n$ for which is guaranteed $T_1^n\left(\frac{1}{x}\right)$ approximates $\frac{1}{x}$ with an error less than $10^{-3}$ on ...
0answers
20 views

### Linear operator exponentials?

A physics book has the following line in it: $$f(k)=e^{k\frac{d}{dx}}f(0).$$ This is, of course, the "correct" Taylor expansion if we write out the series expansion of $e$ and assume that $k$ is a ...
2answers
55 views

### Is it necessary to define $\frac{\sin 0}{0}=1?$ Why not let the Taylor series for $\frac{\sin x}{x}$ determine it? [closed]

Is it necessary to define $$\frac{\sin 0}{0}\overset{\text{def}}{=}1?$$ Why can't we just use $$\frac{\sin x}{x}=1-\frac{x^2}{3!}+\cdots\quad\implies\quad\frac{\sin 0}{0}=1?$$
1answer
34 views

### Taylor Series Expansion to Find Value of Series

How to use Taylor series of $xe^x$ to prove that $\sum_{n=0}^\infty\frac1{(n+2)n!}=1$?
3answers
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### Limit of several variable function using Taylor expansion

I need to find the following limit: $$\lim_{(x,y) \to (0,0)} \frac{x^2+y^2}{1-\cos x\cos y}$$ My approach to find the limit is to first plug in the point, this doesn't work as the function is not ...
6answers
138 views

### Find the approximation of $\sqrt{80}$ with an error $\lt 0.001$

Find the approximation of $\sqrt{80}$ with an error $\lt 0.001$ I thought it could be good to use the function $f: ]-\infty, 81] \rightarrow \mathbb{R}$ given by $f(x) = \sqrt{81-x}$ Because this ...
2answers
77 views

### Are there alternative proofs of the general Taylor-series expansion theorem for real functions?

With a view to better understanding real Taylor series, I have examined some books on basic Calculus, with an eye for the proofs of the Taylor series theorem and the possible authors' comments on its ...
1answer
34 views