# 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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### Problem with Picard Iteration

I have $\frac{dy}{dx} = y^2, y(0) = y_0$ I have solved this as $y = \frac{y_0}{1 - x y_0}$ Which has the Taylor expansion $y_0+y_0^2 x+y_0^3 x^2+y_0^4 x^3+y_0^5 x^4+ ...$ However, when I ...
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### Infinite series expansion of $e^{-x}\cos(x)$

Establish an infinite series expansion for the function $y=e^{-x}\cos(x)$ from just the known series expansions of $e^x$ and $\cos(x)$. Include terms up to the sixth power. I know that the expansions ...
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### What if $f^{(n)}(a)=0$ for all $n\geq 0$?

This morning I was trying to imagine what a function would look like if all it's derivatives were zero at a point $a$ (assuming it is $C^\infty$). My first thought was that it should be identically ...
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### Maclaurin series of $\frac{1}{1+x^2}$

I'm stumped here. I''m supposed to find the Maclaurin series of $\frac1{1+x^2}$, but I'm not sure what to do. I know the general idea: find $\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$. ...
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### Please help me to find Taylor expansion (or approximation) for $f(x)=\frac{1}{x^2(x-1)}$ around $a=2$

First, sorry if my translations is bad. I need help for this exercise, more precisely , I need to know if my result which I've found is good. The exercise: Find Taylor expansion (or approximation) ...
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### Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get decent starting points for some root finding algorithm to the roots of the scaled truncated taylor series of $\exp$. Here W ...
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### Quick way to expand $\cos^{-1}(\cos^2 x)$ up to $O(x^2)$

For a part of a question, I need to expand $\cos^{-1}(\cos^2 x)$ up to $O(x^2)$ about $x=0$. It took me quite a while to get an incorrect answer. What are some quick and efficient offline (i.e, no ...
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### Need help in Taylor series expansion

In this question, I have to write Taylor's series expansion of the function $f(x) = ln(x+n)$ about x = 0, where n ≠ 0 is a known constant. I have done the following: But my professor handed me back ...
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### Laurent series for an infinite sum of functions

I have an infinite sum of analytic functions that is guaranteed to converge for every $x$, except for $x=0$: $$g(x) = \sum_{n=1}^\infty f_n (x)$$ I want to expand the ...
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### Is Fourier series an “inverse” of Taylor series?

I've understood Taylor series as being the representation of a "transcendental" function, using power functions with coefficents represented by appropriate derivatives. (Or maybe it is the MacLauren ...
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### Truncation error using Taylor series

How can we use Taylor series to derive the truncation error of the approximation $$f^\prime(x)\approx\frac{f(x+h)-f(x-h)}{2h}$$
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### Computing the odd terms of the Taylor series of $\frac{z}{e^z-1}$

I know that the terms are $0$ for odd $n > 1$, but I haven't had any luck proving this. Computing them directly verifies this for small $n$; the function is also analytic, so I've tried taking the ...
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### Don't understand why this binomial expansion is not valid for x > 1

today I'm studying binomial expansion and I'm a little confused about when certain expressions are valid. E.g. take this solution from my textbook: I understand that $(1-x)^{-1}$ has an infinite ...
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### How does this Taylor Polynomial work?

The Taylor Polynomial is defined as following: $$P_n(x) = 1 + \dfrac{1}{2}x - \dfrac{1}{8}x^2 + \cdots + (-1)^n \dfrac{1.3.5 \cdots (2n - 3)}{2.4.6 \cdots 2n}x^n$$ If $n = 4$, then the last term in ...
On stackoverflow, a question was asked about getting Mathematica to evaluate the integral, $$\int^\infty_0 \frac{e^{-x}}{\sin x} \, \mathrm{d}x$$ which we know is divergent. In one of the answers, ...
### How can I write $\frac{1}{(a+x)}$ as an exponential function $y = Ce^{-kx}$?
How can I write $\frac{1}{a+x}$, $a$ a non-zero positive constant, in exponential terms in the form of $y = Ce^{-kx}$? I've tried to use to Taylor series but they only seem to work for $x < 1$.