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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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 ...
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Prove the series $\sum_{n=1}^\infty (n(f(\frac{1}{n}) - f(-\frac{1}{n})) - 2f'(0))$ converges

Prove the series $\sum_{n=1}^\infty (n(f(\frac{1}{n}) - f(-\frac{1}{n})) - 2f'(0))$ converges where $f$ is defined on $[-1,1]$ and $f''(x)$ is continuous. I already have a solution for this but I ...
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Taylor series of $F(n) = 1/(n-1)^2 - 1/n^2$ around large n?

I am lost on this Taylor series. The hint says to let $x = 1/n$ and expand around $x = 0$, but I can't make any progress. I am also confused why this hint is helpful. Can't I just expand around ...
Taylor expansion of $\mathbf{u}$ along solutions of $\mathbf{u}' = \mathbf{f}$
Let $\mathbf{u}\colon \mathbb{R}\to \mathbb{R}^{n}$, and suppose $\mathbf{u}$ satisfies $\mathbf{u}' = \mathbf{f}(\mathbf{u}, t)$. To first order, Euler's method says \begin{align*} \mathbf{u}(t+h) = ...