# 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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### Why is the Taylor series of $1/\sqrt{1-4q^2}$ popping up in my recursively defined triangle of polynomials?

While answering this question I stumbled on some nice (inexplicable) observation where a recursively defined sequence of polynomials turned out to coincide with some Taylor development I'll start ...
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### How do you Taylor expand the log likelihood function of the Poisson distribution?

This question is an extension to this previous question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works through the case of a ...
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### Prove the series expansion

Prove that $$(1+x)^\frac{1}{x}=e-\frac{e}{2}x+\frac{11e}{24}x^2-\frac{7e}{16}x^3....$$ where e is exponenial , can any one give a proof...I tried with series expansion i could not get it.
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### Prove the following using Maclaurin's theorem

Prove that $$\log(1+e^x)=\log 2+\frac{1}{2}x+\frac{1}{8}x^2-\frac{1}{192}x^4......$$ I have tried doing it. Tell me if you think the question is wrong
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### How to show $1 +x + x^2/2! + \dots+ x^{2n}/(2n)!$ is positive for $x\in\Bbb{R}$?

How to show $1 + x + \frac{x^2}{2!} + \dots+ \frac{x^{2n}}{(2n)!}$ is positive for $x\in\Bbb{R}$? I realize that it's a part of the Taylor Series expansion of $e^x$ but can't proceed with this ...
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### Taylor series of $\ln(x+2)$

I try to determine the Taylor series of $\ln(x+2)$. Since I know $\ln(1-x) = \sum_{n=1}^{\infty} \frac{x^n}{n}$, I suppose I can rewrite, \begin{align} \ln(x+2) &= ...
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### Taylor series and radius of convergence: $\sqrt{x}$ with centre $x = 16$?

I've been struggling with this question for a while now and getting nowhere with it. Could someone please help me out? Assuming that the function has a power series expansion about the given point, ...
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### Which version of Taylor Theorem is this?

Suppose $X$ is a random variable and $\psi(t)=E[\exp(itX)]$ is its characteristic function. Let $K(t)$ be the principal value of the logarithm of $\psi(t)$. Suppose further that ...
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### Taylor polynomial approximation with absolute error less than 3 decimal places

Assume that f is a function with $|f^{(n)}(x)| \le 11,$ for all n and all real x. Let $T_n(x)$ denote the Taylor polynomial of degree n for f(x) about the point $x=0$. What is the least integer n ...
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### Error on Taylor formula argument

Question: My solution: $$f''(x) = \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$ $$f''(x) = \frac{1}h \frac{f(x+h) - 2f(x) + f(x-h)}h$$ $$f''(x) = \frac{1}{h} [f'(x)-f'(x) = 0]$$ So because the ...
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### If the first nonzero derivative at $a$ is of odd order $n\ge 3$, then $a$ is a point of inflection

Statement to Prove: Let $f$ be a real valued function such that for a fixed point $a$ , $$f^k(a)=0;1\le k\le n-1;\\and\ \ f^n(a)\neq 0.$$ Then if $n$ is odd then $a$ is a point of inflection. ...
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### Second degree multi variable taylor polynomial

Let f (x, y ) = x cos(πy ) − y sin(πx) point: 1,2 I am following the standard formula, which starts with taking the partial of f with regards to x twice, which gives me: ysin(πx)π But plugging in ...
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### Upper Error Bound Taylor Series

(a) Given $f(x) = \sqrt{x}$, find its Taylor polynomial of degree 2 centered at $x=4$ and use it to estimate $\sqrt{5}$. (b) Use Taylor's theorem to give an upper error bound for the estimate in part ...
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### Maclaurin $f(x)=\sin^4x,x\in R$

Write Maclaurin Polynomial$$T\small{10}(x)$$ for function $$f(x)=\sin^4x,x\in R$$ Maclaurin Polynomial: $$T10(x)=f(0)+f'(0)x+f''(0)\frac{x^2}{2!}+...+f^{10}(0)\frac{x^{10}}{10!}$$ For my problem ...
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### The following is a Taylor Series evaluated a particular value of x, find the sum of the series.

This is the Taylor Series in question 1 + $\frac{2}{1!}$+$\frac{4}{2!}$+$\frac{8}{3!}$+...+$\frac{2^n}{n!}$+... I know how to find whether or not the series converges or diverges easily using the ...
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### Evaluating integral using invalid substitution

I was trying to show that for suitable t: $$2\pi(1+t/(\sqrt{(1-t)(3-t)})=\sum_{0}^{\infty}(t^n\int_0^{2\pi}1/(2-cos(\theta))^nd\theta$$ By uniqueness this is clearly the Taylor series about $0$ ...
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### Meaning of $C^k$ in Taylor's expansion [closed]

In the following statement, what does $f \in C^k$ mean? And why is there a $q$ for the last part of expansion? So now if I let $k = 2$, what does it mean? And will the expansion involve 3nd ...
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### Remainder of Taylor approximation

Consider the ODE $\dot{x}=f(x)$ with $f(x)$ smooth and let $x_0$ be an equilibrium, i.e. $x(t)=x_0=\text{const}$ and $f(x_0)=0$. The substitution $x=x_0+y$ shifts the origin to $x_0$. With the new ...
### A funny question: Taylor polynomials and series associated with the Lost numbers $4, 8, 15, 16, 23, 42$
The interpolation polynomial for the "Lost" numbers $4, 8, 15, 16, 23, 42$ is $$P(x)=60-\frac{612}{5}x+\frac{367}{4}x^{2}-\frac{235}{8}x^{3}+\frac{17}{4}x^{4}-\frac{9}{40}x^{5}.$$ That is, $P(1)=4$, ...