# 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.

5,580 questions
Filter by
Sorted by
Tagged with
7answers
146 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 ...
0answers
29 views

3answers
44 views

2answers
56 views

### 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 ...
1answer
76 views

1answer
53 views

2answers
256 views

2answers
55 views

### 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 ...
0answers
28 views

### 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) = ...
2answers
85 views

### Applying a Taylor series “with respect to $a/r$” and “around $0$”

My question is what does it mean applying a Taylor series with respect to something and around a point. What is the difference? Please explain it with the following example: Apply a Taylor ...