# 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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### Consider the function $f(x) = e^{x^2}\ln(1+x)$ for $0 < x < 1$

So I was able to do the first half of this problem (part a), which was: $$e^{x^2}\ln(1+x) \approx x - \frac{x^2}{2} + \frac{4x^3}{3}$$ but I'm confused what my next step should be, for solving (part ...
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I know that for a (finite) polynomial $P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_0$ whose zeros are $x_1, x_2, \ldots, x_n$, then we can factorize it as $$P(x) = a_n(x - x_1)(x - x_2) \cdots (x -... 2answers 35 views ### Can we Relate Radius of Convergence of Taylor Series and Asymptotic Rate of Growth? I still need to be disabused of the belief that there is some simple connection between the finiteness of the radius of convergence and the asymptotic rate of growth. 1. Can we develop any ... 1answer 20 views ### Is Every (Real) Analytic Function (with Non-Degenerate MacLaurin Series) Asymptotically Greater Than any Polynomial? Question: Given a function f: \mathbb{R} \to \mathbb{R} such that the MacLaurin series exists and equals the function for every x \in \mathbb{R}, and such that for all n \ge n_0, n_0 some ... 0answers 36 views ### Taylor series Lagrange Remainder explanation So, given a Taylor series:$$f(x)=f(x_0)+f'(x_0)(x-x_0)+f''(x_0)\frac{(x-x_0)^2}{2!}+\cdot\cdot\cdot+f^{(n)}(x_0)\frac{(x-x_0)^n}{n!}+R_n$$The error R_n is given by:$$R_n=\frac{f^{(n+1)}(\xi)}{(n+...
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Find three nonzero terms of the Maclaurin series of the function $f(x)={3/5} tan5x/x$ Using the maclaurin series i found them to be.. $3/5+x^2/25+2x^4/25$ Is this correct? If not what is the ...
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### Could some confirm my answer for this limit using taylor series?

$\lim_{x→0}$ $\dfrac{x^2}{x\sqrt{1+x} −\ln(1+x)}= ?$ I got $-2$. Is this correct if not what is the answer so i can find out where i went wrong. Thanks in advance
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### Find a function f so that Taylor expansion is always accurate to this degree

Find a function $f$ from R to N such that with $T$ be the Taylor expansion of $\sin(x)$ around $0$. $| \sin (x) - T_{f(x)}x$| $\leq 1$ The hint is to use $n! \leq 3 \sqrt{n} {(\frac{n}{e})}^n$
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### Taylor series question help!

This question is on a past paper for my exam but no model solutions have been provided and I'm worried I'm doing completely the wrong thing, Consider two functions represented by Taylor (MacLaurin) ...
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### Limit calculate using Maclaurin series

I need help to calculate this limit using Maclaurin series: $\lim_{x\to \infty}((x^3-x^2+\frac{2}{x})e^{\frac{1}{x}}-\sqrt{x^3+x^6})$ I don't know from where to start. I think I need to to write ...
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### Taylor Maclaurin series

Can someone explain to me how this equals? I'm taking a calculus III course at the moment, and I'm doing Taylor and Maclaurin series at the moment, and this is the last step of a problem, but i don't ...
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### Big 'O' Notation - Taylor Series

Q) Use the Taylor Series Expansion to show the first derivative f '(x) can be approximated by $$-(3f(x) -2f(x+h) - f(x-h) / h )$$ What is the precision? Now I found after using the Taylor ...
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