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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2answers
40 views

Show that $\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1$

Show that $$\left\vert\frac{\pi}{4} - \left(1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9}\right)\right\vert < 0.1 .$$ I know that $\arctan 1 = \frac{\pi}{4}$ and that the sequence ...
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1answer
16 views

Laurent series 1/(z+2)+1/z^2, 0<|z+2|<2

I can't find a way to represent $z^2$ in terms of $z+2$. I've tried to do $(z+2)(z-2)+4$, but I'm stuck with the $+4$.
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0answers
13 views

Relative position of a function's surface and a tangent plane

Let $f$ be a function such that \begin{align*} f(x,y)=&f(x_0,y_0)+(x-x_0)\frac{\partial f}{\partial x}(x_0,y_0)+(y-y_0)\frac{\partial f}{\partial y}(x_0,y_0)+ \\ &\frac{1}{2}\left[ (x-a)^2\...
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0answers
21 views

Laurent Expansion of a Composition (with the inverse)

Suppose $f(z) = a_1z + a_0 + O(\frac{1}{z})$ as $|z| \rightarrow \infty$. Here $a_1 > 0$. Let $W(z) = z + \frac{1}{z}$. How do I obtain that the composite function $g(z) = W \circ f^{-1}(z)$ has ...
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2answers
42 views

Finding Taylor Series of the following function?

How do I go about finding the answer to this? I have tried two methods, using the known taylor series for $\sin (x)$ and $\frac{1}{1-r}$ and then transform and add those, and I have also tried by ...
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0answers
34 views

Is it possible to compute this infinite series explicitly? [on hold]

Is it possible to compute explicitly or in an approximate way the infinite series: $$S(x)=\sum_{n=0}^{\infty}\frac{a^{n}}{n!}f^{(2n+1)}(x),$$ for a generic function $f$?
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4answers
62 views

a hint for this Taylor series$ \frac{\cos\left(2x\right)-1}{x^2}$

Compute the first three terms (nonzero) $\frac{\cos\left(2x\right)-1}{x^2}$ the first term is $\cos \left(2\right)-1$ but in the answer, the first term that I have to choose is... $-2$ or $2$ or $-1/...
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1answer
73 views

Find the Laurent series expansion of $\frac{z}{(1-z)^2}$ [on hold]

I want to find the $g(z) = \frac{z}{(1-z)^2}$ for $z \in \mathbb{C}-\{1\}$. My approach has been to decompose the fraction, i.e. $$\frac{z}{(1-z)^2} = \frac{A}{1-z} + \frac{B}{(1-z)^2} \implies A(1-z) ...
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1answer
24 views

How to prove finite difference approximation has error of order $\mathcal{O}(\Delta x^2)$

I'm asked to prove that the finite difference approximation $$u_{xx}(x_i) = \frac{u_{i+1} - 2u_i + u_{i-1}}{\Delta x^2}$$ gives a discretization error of order $\mathcal{O}(\Delta x^2).$ My attempt: ...
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0answers
18 views

Taylor Expansion and Identities in Limits— when to use what? [on hold]

How will I know when to use Taylor series and when to use identities in a sum involving limits?
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0answers
24 views

For function $f(\frac{1}{n})=\frac{n^2}{n^2+1}$ calculate $f^{(k)}(0)$ for every $k=0,1,2,…$ [duplicate]

For function $f: \mathbf{R} ->\mathbf{R}$ which is arbitrarily many times derivative function and for every $n \in \mathbf{N}$ we have $$f(\frac{1}{n})=\frac{n^2}{ n^2+1}$$ Calculate $f^{(k)}(0)$ ...
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1answer
48 views

Taylor series of $\ln(1+x+x^2+…+x^{10})$

I have to find Taylor series of $\ln(1+x+x^2+...+x^{10})$. I have a clue that I write like the two difference of logarithm but I do not how. Any help?
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5answers
68 views

Taylor series of $(1+3x) \cdot \ln(1+x)$

I have to find Taylor series of $(1+3x) \cdot \ln(1+x)$. I know Taylor series of $(1+3x) \cdot \ln(1+x)$ but I do not know hot to simplify. Any help?
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1answer
39 views

If $f$ is twice-differentiable at $a$, show that if $f''(a)>0$ then $f(a)+f'(a)(x-a)\leq f(x)$ in a region of $a$.

Given some function $f: I \subseteq\mathbb R \rightarrow \mathbb R$, Which is differentiable twice at some point $a\in I$. Prove: If $f''(a)>0$ then, $f(a)+f'(a)(x-a)\leq f(x)$ in a ...
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0answers
47 views

How do I evaluate the antiderivative of $e^{cos(x)}$?

Functions that do not have an elementary antiderivative can be evaluated by generating a Taylor series, provided the function is infinitely differentiable and uniformly convergent in its domain. ...
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1answer
21 views

linearisation of a system of differential equations (first order Taylor method)

The first order Taylor expansion of this system of equations in the picture if pretty straightforward. However I don't get how we get values for the steady state. Is it correct to say that in the ...
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2answers
35 views

How to treat absolute value bars in $\log|\frac{1+x}{1-x}|$ to show is even function to solve complex integral?

How to treat absolute value bars in $\log\left|\frac{1+x}{1-x}\right|$ to show is an even function? When dealing with strictly numbers, one defines the absolute value of $x$ as: $$|x| =\begin{cases} ...
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1answer
53 views

When $|f(z)| \leq \frac{1}{\sqrt{1-|z|^2}}$ on open unit disk $\mathbb{D}$ then $|f'(0)| \leq 2$

I have to show, that if $|f(z)| \leq \frac{1}{\sqrt{1-|z|^2}}$ on the open unit disk $\mathbb{D}$ then $|f'(0)| \leq 2$. I thought I could use Cauchy's estimates theorem ,where $\left|f(z)\right| \le ...
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1answer
26 views

Minimum value of $n$ for Lagrange reminder on Taylor polynomial of $\frac{1}{x}$

I'm trying to solve the following question: "Find the minimum value of $n$ for which is guaranteed $T_1^n\left(\frac{1}{x}\right)$ approximates $\frac{1}{x}$ with an error less than $10^{-3}$ on ...
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0answers
20 views

Linear operator exponentials?

A physics book has the following line in it: $$ f(k)=e^{k\frac{d}{dx}}f(0). $$ This is, of course, the "correct" Taylor expansion if we write out the series expansion of $e$ and assume that $k$ is a ...
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2answers
55 views

Is it necessary to define $\frac{\sin 0}{0}=1?$ Why not let the Taylor series for $\frac{\sin x}{x}$ determine it? [closed]

Is it necessary to define $$\frac{\sin 0}{0}\overset{\text{def}}{=}1?$$ Why can't we just use $$\frac{\sin x}{x}=1-\frac{x^2}{3!}+\cdots\quad\implies\quad\frac{\sin 0}{0}=1?$$
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1answer
34 views

Taylor Series Expansion to Find Value of Series

How to use Taylor series of $xe^x$ to prove that $\sum_{n=0}^\infty\frac1{(n+2)n!}=1$?
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3answers
49 views

Limit of several variable function using Taylor expansion

I need to find the following limit: $$ \lim_{(x,y) \to (0,0)} \frac{x^2+y^2}{1-\cos x\cos y}$$ My approach to find the limit is to first plug in the point, this doesn't work as the function is not ...
4
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6answers
138 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 ...
4
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2answers
77 views

Are there alternative proofs of the general Taylor-series expansion theorem for real functions?

With a view to better understanding real Taylor series, I have examined some books on basic Calculus, with an eye for the proofs of the Taylor series theorem and the possible authors' comments on its ...
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1answer
34 views

Find the Taylor series for $f$ at $t= 0$ where $f(t) = a + \int_0^tsin(t-s)f(s)ds$.

Let $a, b\in \mathbb{R}$ and $b\notin 0$. Suppose that there is exist $f\in C([-b, b])$ such that, for all $t\in [-b,b]$, $$f(t) = a + \int_0^tsin(t-s)f(s)ds$$ 1- Show that if $f$ exist, then $f\in ...
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2answers
256 views

Prove this matrix to be unitary

This is a homework question so, hints are appreciated. But if someone is generous enough, to show the full calculation, I'd be quite grateful! Say a matrix B is anti-hermitian:$$\begin{bmatrix} i &...
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1answer
47 views

Lipschitz condition for Frobenius norm for Complex matrices

For the function $f({\bf x}) = ||{\bf Ax - b}||^2_2$, where the vectors are ${\bf x} \in \mathbb{C}^{n \times 1}, {\bf y} \in \mathbb{C}^{m \times 1}$ and martix ${\bf A} \in \mathbb{C}^{m \times n}$,...
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1answer
55 views

Verify the matrix exponential $e^{i\hat{H}t/\hbar}$ is unitary

What could be an example of an anti-hermitian matrix $i\hat{H}t$ , which satisfies the matrix exponential $$e^{i\hat{H}t/\hbar}$$ being a unitary matrix?
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1answer
63 views

Complexity of calculating $f^{(n)}(0)$/extracting a coefficient of a functions taylor-series

Many combinatorial problems can be solved using generating functions. In such a case, we obtain a function $f(x)$, which (for usual) has a taylor-expansion: $$ f(x) = \sum_{n\ge 0 } a_n x^n $$ So ...
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1answer
110 views

Second order Taylor expansion of Frobenius norm

I have the following function $||{\bf A} - {\bf BC}||^2_F$, where ${\bf A} \in \mathbb{C}^{m \times n}$, ${\bf B} \in \mathbb{C}^{m \times k}$, and ${\bf C} \in \mathbb{C}^{k \times n}$, which is a ...
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0answers
74 views

Taylor expansion of Frobenius Norm

I have the following function $||{\bf A} - {\bf BC}||^2_F$, where ${\bf A} \in \mathbb{C}^{m \times n}$, ${\bf B} \in \mathbb{C}^{m \times k}$, and ${\bf C} \in \mathbb{C}^{k \times n}$, which is a ...
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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) = ...
1
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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 ...
3
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1answer
104 views

Find $f(x) = x + \frac{2}{3}x^3 + \frac{2\cdot4}{3\cdot5}x^5 + \frac{2\cdot4\cdot6}{3\cdot5\cdot7}x^7+\cdots$ where $|x|<1$ [duplicate]

Find $$f(x) = x + \frac{2}{3}x^3 + \frac{2\cdot4}{3\cdot5}x^5 + \frac{2\cdot4\cdot6}{3\cdot5\cdot7}x^7+\cdots +\infty\,,\quad|x|<1$$ My solution: $$f'(x) = 1 + 2x^2 + \frac{2}{3}\cdot4x^4 + \frac{...
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0answers
37 views

Finding a third-degree approximation of $x^3y'' + x^2y' - (e^x-1)y=0$

I need to find a third-degree approximation of $x^3y'' + x^2y' - (e^x-1)y=0$ at some singular point. I've found that the singular points of $x^3y'' + x^2y' - (e^x-1)y=0$ are 0 and $\infty$. However, ...
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1answer
28 views

How to decide if a DE has a singular point at infinity, and if so, what to do about it.

I'm trying to understand singular points at infinity, so looking at the example $$x^2(1-x^2)y'' - y = 0$$ I am trying to investigate (a) does it have a singular point at infinity, and (b) if it ...
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3answers
60 views

show that $|\sin(1)-0.841|\leq10^{-3}$

My textbook states that $|\sin(1)-0.841|\leq10^{-3}$ but I do not know how this could be true. It also gives a table showing values of ${1\over n!}\pm10^{-8}$ when $n$ is $1$ to $10$. The reason why ...
0
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2answers
67 views

How to get the result of only first N terms of a geometric series like $1+Ax+Ax^2+Ax^3+Ax^4+Ax^5$… [duplicate]

Given -1 < x < 1, and for Series like the following, I am trying to figure out not the complete Total, but only the Total of first N Terms. So the Question is: What is the Total of first N ...
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3answers
76 views

What is $1+Ax+Ax^2+Ax^3+Ax^4+Ax^5$… an expansion of ?? [closed]

$$1 + x+ x^2 + x^3 + x^4+ x^5 + x^6...$$ is the expansion of $\frac{1}{1-x}, $ and $$1 - x + x^2 - x^3 + x^4 - x^5 + x^6...$$ is the expansion of $\frac{1}{1+x}.$ I am trying to figure out of what ...
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1answer
45 views

Is really this :$\int_{0}^{t}\operatorname{erf}(x+\sqrt{1-\log (x)} )dx \sim t$ true for every $t$?

I have accrossed this integral when I run some of my computation in Wolfram alpha with many values of $t$ , Really seems to conjecture that : $$\int_{0}^{t}\operatorname{erf}(x+\sqrt{1-\log (x)} )dx ...
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1answer
16 views

How to use linearization at the point where the given function is not defined

Let's say we are given a function $f(x)$, which is not defined at the point $x_0$. How do we find linear approximation of $f$ near $x_0$? P.S. I wrote "linear" just to make things simpler, I came ...
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2answers
63 views

Why Using Maclaurin series is giving me a different answer?

$$\lim_{x \to\infty }\frac{(1+\frac{1}{x})^{x^{2}}}{e^{x}}= \frac{1}{\sqrt{e}}$$ I have to proove this equation using Maclaurin expansion, which I know how to do. However, my question is when looking ...
1
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1answer
50 views

Show that there are no entire functions such that $\bigcup_{n = 0}^{\infty} \{ z \in \mathbb{C} : f^{(n)}(z) = 0 \} = \mathbb{R}$.

Show that there are no entire functions such that $\bigcup_{n = 0}^{\infty} \{ z \in \mathbb{C} : f^{(n)}(z) = 0 \} = \mathbb{R}$. My attempt: So I tried this by contradiction. Suppose there is an ...
0
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2answers
66 views

Taylor series expansion of $x^x$ [closed]

I am well aware of the expansion by using $e^{x*ln{x}}$ and am looking for a different way. Can somebody please tell me a different expansion for $x^x$?
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0answers
38 views

Numerators of Maclaurin series coefficients

I have noticed that often the Maclaurin series of notable functions have rational coefficients whose denominators are relatively easy to understand, while the numerators are intractable. Two examples ...
1
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1answer
55 views

How to show that the remainder of this Taylor expansion of this homogeneous function is zero?

In Calculus of Several Variables, Third Edition, by Serge Lang, this is exercise 1 in chapter 6, section 5: Let $f$ be a function of two variables. Assume that $f(O) = 0$, and also that $f(t P) = t^...
1
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1answer
30 views

Composition of series and Taylor expansion

Let's say $p_n (z)$ is the Taylor-expansion of a function $a(z)$ up to the $n$-th order. (Consider $a:\mathbb{R} \rightarrow \mathbb{R}$). Now I have a series $x_n\rightarrow x$ for $n \rightarrow\...
0
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0answers
27 views

Multivariate finite difference formulas

Consider a Taylor expansion of a function $f$ of $N$ variables $\mathbf{x}$, about $\mathbf{x}=\mathbf{0}$: $$ f(\mathbf{x}) = f(\mathbf{0})+\sum_i^Nc_ix_i + \frac{1}{2!} \sum_{i,j}^N c_{ij}x_ix_j + \...
1
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2answers
67 views

First Order Approximation of a $\sqrt{3.9}$

Use the first order approximation to determine $\sqrt{3.9}$. Use your calculator to determine the error in your approximation. I am really confused by this question. There is no function so I don't ...