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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4
votes
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 ...
1
vote
0answers
29 views

Good asymptotic for this recursion?

Consider some initial integer values and let the integer sequence continue like $$\begin{align}f(n) &= f(n-1) \\ &+ n(n+1)(n+2)\dots(n+5) f(n-2) \\ &- \frac{ n(n+1)(n+2)\dots(n+5)}{2} f(...
0
votes
1answer
34 views

How to prove this recursion for these Taylor coefficients?

Consider the Taylor series $$ f(x) = \frac{1 - 2x - \sqrt{1 - 8 x + 8 x^2}}{2x(1-x)} = f_0 + f_1 x + f_2 x^2 + ... $$ It appears that the Taylor coëfficiënts start to follow the recursion $$ 0 = (...
0
votes
3answers
44 views

Find an equivalent sequence

Consider $ u_n = (n+1)^{1/n+1} - n^{1/n} $ Find an equivalent sequence at infinity. (meaning $ u_n / y_n \rightarrow 1 ) $ I tried doing : $ u_n = e^{ \frac{ln(n+1)}{n+1}}(1 - e^{\frac{ln(n)}{n} -...
1
vote
5answers
33 views

Maclaurin expansion of $\arccos(1-2x^2)$

Maclaurin expansion of $\arccos(1-2x^2)$ This is what I tried. $f'(x)=2(1-x^2)^{-1/2} \\ f''(x)=2(1-x^2)^{-3/2}+3 \cdot 2 x^2(1-x^2)^{-5/2} \\ f^{(3)}(x)=18x(1-x^2)^{-5/2}+2\cdot 3\cdot 5x^3(1-x^2)^{...
0
votes
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 ...
0
votes
1answer
76 views

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

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) ...
1
vote
2answers
30 views

Laurent series of $~\frac{1}{z+2}+\frac{1}{z^2}, ~~~~~~~0<|z+2|<2~$

Laurent series of $$~\frac{1}{z+2}+\frac{1}{z^2}, ~~~~~~~0<|z+2|<2~$$ 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 $+4$.
4
votes
4answers
112 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 ...
3
votes
1answer
666 views

Analytic continuation of ln(z) counterclockwise about the unit circle,

We write ln(z) as ln(1+z-1) = ln(1+(z-1)) to utilize the familiar expansion that is: (z-1) - (z-1)^2 / 2 + ... which converges for |z-1| < 1, i.e., we get convergence of ln(z) in an open Taylor ...
0
votes
0answers
17 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\...
4
votes
3answers
95 views

Combinatorial Proof that the Logarithm of a Product is the Sum of the Logarithms

I've been strongly drawn recently to the matter of the fundamental definition of the exponential function, & how it connects with its properties such as the exponential of a sum being the product ...
2
votes
1answer
2k views

Trapezoid rule error

I am trying to compute the error in the trapezoid rule integration for a function $f(x)$ in the interval $[a,b]$. I believe I have to Taylor-expand $f(x)$ around $x=a$ $f(a) + (x-a)f'(a)+ 1/2 (x-a)...
3
votes
1answer
105 views

Determine the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$

I need to calculate the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$ . It seems very "similar" to Taylor expansion of functions arcsin(x) and its derivative for x = -2. It is known: $...
0
votes
2answers
43 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 ...
2
votes
0answers
25 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 ...
0
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0answers
36 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 $f$?
1
vote
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/...
0
votes
1answer
25 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: ...
5
votes
3answers
213 views

computing the series $\sum_{n=1}^\infty \frac{1}{n^2 2^n}$

$$\sum_{n=1}^\infty \frac{1}{n^2 2^n}$$ I am new in series thus I tried a pair of methods to compute but I couldn't
57
votes
11answers
42k views

Simplest proof of Taylor's theorem

I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem. By which I mean this: there are plenty of proofs that introduce some arbitrary construct: ...
5
votes
5answers
69 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?
-1
votes
0answers
19 views

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

How will I know when to use Taylor series and when to use identities in a sum involving limits?
0
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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 $f^{(k)}(0)$ ...
1
vote
1answer
51 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?
0
votes
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 ...
2
votes
0answers
50 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. ...
0
votes
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 ...
0
votes
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 ...
1
vote
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} ...
1
vote
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 ...
0
votes
1answer
45 views

Strange behavior of composite MacLaurin Series

While answering a question about the MacLaurin Series Expansion of a composite function I noticed something strange I can not explain to myself. The task was to verify that the MacLaurin Series ...
1
vote
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 ...
2
votes
2answers
57 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 $$\frac{\sin x}{x}=1-\frac{x^2}{3!}+\cdots\quad\implies\quad\frac{\sin 0}{0}=1?$$
0
votes
1answer
34 views

Taylor Series Expansion to Find Value of Series

How to use Taylor series of $xe^x$ to prove that $\sum_{n=0}^\infty\frac1{(n+2)n!}=1$?
0
votes
1answer
48 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 ${\bf A} \in \mathbb{C}^{m \times n}$,...
4
votes
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 ...
1
vote
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 ...
0
votes
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 ...
4
votes
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 &...
23
votes
2answers
285k views

taylor series of $\ln(1+x)$?

Compute the taylor series of $\ln(1+x)$ I've first computed derivatives (up to the 4th) of ln(1+x) $f^{'}(x)$ = $\frac{1}{1+x}$ $f^{''}(x) = \frac{-1}{(1+x)^2}$ $f^{'''}(x) = \frac{2}{(1+x)^3}$ $f^...
1
vote
1answer
64 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 ...
1
vote
1answer
56 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?
0
votes
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 ...
0
votes
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 ...
2
votes
1answer
251 views

Using Taylor's theorem and Lagrange form of the reminder to prove the second order condition for convexity

I try to prove the second order condition for convexity. So far' I've done the following: First, I prove second order => convexity: Let $f$ be a function with positive semi-definite Hessian. Using ...
0
votes
1answer
43 views

$f(x)= \tan x; f^n(0)-{n\choose 2}f^{n-2}(0)+ {n\choose 4}f^{n-4}(0)+…=\sin\frac{n\pi}{2}$

How should I go about proving following relation $f(x)=\tan(x)$, then $$f^n(0)-{n\choose 2}f^{n-2}(0)+ {n\choose 4}f^{n-4}(0)+...=\sin\frac{n\pi}{2}$$ I tried Maclaurin expansion but not able to get ...
-1
votes
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 ...
0
votes
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
vote
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 ...