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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19 views

Maclaurin series - Approximation and interval of convergence

This is a problem which I should apparently be solving with Maclaurin series, but I failed to do so. So I attempted it with binomial series, with 5 terms and an error less than the requirement in ...
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1answer
17 views

Taylor series of an analytic function that maps the unit disk surjectively onto the upper half plane

Given only that $f(z)$ is analytic and maps the unit disk $|z| < 1$ surjectively to the upper half plane $\Im(z) > 0$, how much can we deduce about $f(z)$? In particular, can we find the radius ...
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0answers
36 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
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1answer
24 views

Lagrange remainder vs. Alternating Series Estimation Theorem: do they always give you the same error bound?

Given a function and its nth degree Taylor series approximation, we can use the Lagrange form of the remainder to get a maximum value of the error of approximation. If the series is also an ...
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68 views

A function equal to its Taylor series on an interval is also equal to its Taylor series on a subinterval with different center

Suppse the power series $ \sum_{n=0}^\infty a_n (x-a )^n$ has positive radius of convergence $R$ and thus defines a real analytic fuction $f$ on $(a-R,a+R).$If $x_0$ is a point with $|x_0-a|<R,$ ...
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2answers
64 views

Taylor series $(x+1)^{\frac{1}{3}}$

Complete the Maclaurin polynomial of degree three for $(x+1)^{1/3}$. I have completed the first two derivatives of this function and thus have coefficients but am not certain how to put them into ...
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2answers
40 views

Convergence of general Taylor series

For any $a,h \in \mathbb{R}$, how can we see the series, $\sum_{k=0}^{\infty} \dfrac{f^{(k)}(a)}{k!} h^k$ converges to $f(a+h)$?
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40 views

When to Taylor expand in a differential equation

I've come across a differential equation: $\dfrac{d\theta}{dx} = f(\theta)$, whose analytic solution is very complicated, but in this situation it is valid to Taylor expand functions around $\theta ...
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93 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
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41 views

Estimate eigenvectors of symmetric matrix with almost vanishing diagonal

Is there a way to approximate the eigenvectors of a symmetric matrix with almost vanishing diagonal elements, i.e. with the block matrix form, \begin{equation} M=\left( \begin{array}{cc} ...
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11 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...
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1answer
25 views

Taylor's Theorem in 2 variables

I've never been 100% happy with using taylors theorem, mainly because I see it used in a bunch of different ways and I'm never sure in which situations it is valid. The way I was introduced to it was ...
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2answers
29 views

Evaluating a limit with Taylor Series

I would like to find the following limit using Taylor Series: $$\lim_{x\to0}\frac{6\sinh x-6x-x^3}{x^4(6+x^2)\sinh x}.$$ Now my question is the following: How do I know exactly how to approximate ...
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13 views

Lagrange's form of the remainder vs Cauchy's form

So far (while practicing exercises) I've used Lagrange's form of the remainder. Is there a situation when Cauchy's form comes in handy while Lagrange's form fails for some reason? Is there a rule of ...
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62 views

How to express $e^{yS^2}(f(x))$ in closed form where $\frac{d}{dx}=S$

$$ f(x)+\frac{y.f'(x)}{1!}+\frac{y^2 f^{''}(x)}{2!}+\cdots=e^{yS}(f(x))=f(x+y) \text{ where }\frac{d}{dx}=S$$ is a operator $$ f(x)+\frac{y.f''(x)}{1!}+\frac{y^2 ...
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72 views
+50

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
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50 views

Given a function $f$, find the largest $n$ such that $f(x)/x^n$ can be defined at $x=0$ to become differentiable there

Let $f(x) = \ln\left(\frac{x^2}{2}+1\right)+\cos x -1$. Find the largest $n\in\Bbb{N}$ such that there is $C\in\Bbb{R}$ such that: $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C ...
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1answer
21 views

Manipulating Taylor series with $\operatorname{Log} z$

I'm trying to prove the following: $$ \operatorname{Log}z = \sum_{n=1}^{\infty} \frac{\left(1-\frac1z\right)^n}{n} $$ for $\left|1-\frac1z\right|<1$. Does anyone have advice on where to ...
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1answer
24 views

Laurent series of $f(z)=(z^2-1)\mathrm{cos}\frac{1}{z+i}$ in $z_{0}=-i$ and $Res[f(z), -i]$

This is how I've done so far: $$ f(z)=(z^2-1)\mathrm{cos}\frac{1}{z+i} \\w=z+i \;\;\;\; \Rightarrow \;\;\;\; z=w-i \\f(w)=((w-1)^2-1)\mathrm{cos}\frac{1}{w}=(w^2-2wi-2)\mathrm{cos}\frac{1}{w} ...
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1answer
29 views

lim and Maclaurin

I have this assingment: $$\lim_{x\to0}\frac{\ln(1+x^2)-\ln(1-x^2)}{e^{x^2}-e^{-x^2}}$$ And by using the Maclaurin approximations I get this: ...
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55 views

Show $\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$

I'm reading a proof which aim to show that: $$\forall x>0:\ln(1+x) > x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$$ the Taylor expansion of $\ln(1+x)$ is (not by chance): $$x - ...
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32 views

Taylor expansions in two variables

I need help in this proof Can I use Taylor expansion in Algebra? someone could give more detail of this Taylor expansion? Thanks in advance EDIT The main question is how the author get this "Y + ...
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1answer
36 views

Solution of $d^2u/dx^2 + u/A = 0 \ (\text{or } \ C),$ with conditions

Does the following ODE: $$d^2u/dx^2 + u/A = 0 \quad (\text{or } \ C),$$ have a solution with the conditions: $$ \left.\frac{d^2u}{dx^2}\right|_{x=0} = 0, $$ $$u(x=0) = B$$ and $$ ...
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49 views

Evaluate a limit using Taylor series

Let $$\lim\limits_{x\to 0}\frac{({\ln(1+x) -x +\frac{x^2}{2})^4}}{(\cos(x)-1+\frac{x^2}{2})^3}$$ Now, I know that I should utilize Taylor polynomial. $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ...
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2answers
37 views

Multplication of series

My textbook is taking about the Cauchy product and I don't quite understand it and it says that when multiplying series, the sum of the third one is equal to the product of the sums of first two ...
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21 views

Relationship between Lagrange interpolation and Taylor expansion

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h_{-1}$ and $x_{1} = x_0 + h_1$ with $h_1, h_{-1}$ > 0. Given a smooth function f, and an approximation to $f'(x_0)$ given by the ...
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1answer
25 views

Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
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1answer
48 views

Taylor series and Lagrange's remainder f(x)=$e^x$

In my textbook the Lagrange's remainder which is associated with the Taylor's formula is defined as: $R_{n}(x)= \frac{(x-a)^n}{n!} f^{(n)}(a + \vartheta (x-a))$, for some $\vartheta$ $\in$ <0 ,1> ...
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72 views

Relationship between $\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$ and $\sum\limits_{n=0}^\infty \frac{a_n^2 x^n}{n!}$

For an analytic function with the property $f^{(n)}(0)=a_n$, we have $f(x)=\sum\limits_{n=0}^\infty \frac{a_n x^n}{n!}$. This can be extended to $f^{(n)}(x)=\sum\limits_{n=0}^\infty \frac{a_{n+1} ...
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1answer
29 views

How to find the first $4$ nonzero terms of a Taylor series at $x=0$ [closed]

I'm having trouble solving this one. All detailed work is helpful. Find the first four non zeros of the formula: $$f(x)= \frac{4}{(1-x)^3} \quad\text{at}\quad x=0$$ $$f(x)=\frac{x^5}{1-x} ...
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39 views

Approximation for lower incomplete gamma function

Does any one know approximations for the lower incomplete gamma function $\gamma(a,bx)$. The problem is this: I want to find the quantile function for the CDF of the gamma distribution. The CDF of the ...
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1answer
58 views

Calculating Laurent Series of Complex Function

How does one alternate the Bernoulli number series expansion $$\frac x{e^x - 1}=\sum_{n=0}^{\infty}\frac{B_nx^n}{n!}$$ To calculate the Laurent Series centered at 0 in the annulus of convergence of ...
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30 views

Laurent Series expansion with |z-1|

Here's the problem: Expand $\dfrac{e^z}{z-1}$ in a Laurent series convergent in $0 < |z-1| < \infty$. And here is my attempt, I just wanna know if it's right cause it seems kinda simple. ...
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15 views

Proving Taylor Series Estimation of Integrals

Given that $f, g, h$ are all continuos on $x \in [a,b]$, $g(x)$ approximates $f(x)$ with an error of at most $h(x)$ Meaning $|f(x) - g(x)| \le h(x)$ Have to prove $|\int_a^bf(x)\,dx - ...
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1answer
27 views

How many terms of the Taylor expansion should I develop?

How do I know how many terms of the power series should I develop to evaluate a limit? Example: Given this limit: $L:=\displaystyle\lim_{x\to0}\left(\frac{\ln(1+x)}x-e^{-x/2}\right)\frac1{\cosh x - ...
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2answers
38 views

Two-dimensional Taylor linearisation

I have to perform a first order taylor expansion of a function $f(\vec{x}) = f(x+u,y+1)$ at the point $\vec{a} =(x,y)$. My solution reads $$ f(\vec{x}) \approx f(x,y) + \left( \begin{matrix} ...
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1answer
55 views

Showing Taylor Series for $f(x) = e^{-x^2}$ converges to $f$

Show Taylor Series for $f(x) = e^{-x^2}$ converges to $f$ I am stuck because when taking the (n+1) th derivative of f, I do not see a general pattern. Meaning I am having difficulty in bounding ...
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17 views

Showing uniform continuity of function giving radius of convergence

Let $f$ be an analytic function on an open disk $D$ and let $R(z)$ denote the radius of convergence of the power series of $f$ about a point $z$. Is there an easy way to show that $|R(z_1) - R(z_2)| ...
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29 views

PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial ...
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1answer
55 views

Taylor Series Theorem

So I see the argument presented in taylor series, that $$\sum c_n (x-a)^n = \sum \frac{f^{(n)}(a)}{n!} (x-a)^n$$ or $c_n = f^{(n)}(a)/n!$ if $x=a$ the question is, since the above only holds when ...
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55 views

Finding $\lim_{x\to0} \frac{\sin(x^2)}{\sin^2(x)}$ with Taylor series

Evaluate $$\lim_{x\to0} \frac{\sin(x^2)}{\sin^2(x)}.$$ Using L'Hospital twice, I found this limit to be $1$. However, since the Taylor series expansions of $\sin(x^2)$ and $\sin^2(x)$ tell us that ...
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1answer
47 views

Function whose power series coefficients contain logarithms

Is there a function that can be expressed as a power series $$f(x)=\sum_{n=0}^\infty a_n x^n$$ whose coefficients $a_n$ are expressions containing $\log n$ or something similar?
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29 views

Estimating the remainder of Taylor series written in Lagrange form

Given the function $$f(x) = \ln\left(\frac{1+x}{1-x}\right)$$ Show that the error $f(1/3) - T_n(1/3)$ is at most $55/7776$ My attempt Remainder Term = $[f^{(5)}(x) = 24/(1+x)^5 + ...
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605 views

Why does this infinite series equal one?

Why does $$\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$$ Is there an intuitive method by which to derive this equality?
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2answers
79 views

taylor series of ln(1+x)?

Compute the taylor series of $ln(1+x)$ I've first computed derivatives (upto the 4th) of ln(1+x) $f^{'}(x)$ = $\frac{1}{1+x}$ $f^{''}(x) = \frac{-1}{(1+x)^2}$ $f^{'''}(x) = \frac{2}{(1+x)^3}$ ...
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28 views

Tangent Line Approximation Using Taylor Series

I am new to Taylor series as a whole and was wondering if someone with a bit more background could validate my thought process in answering the following question. Question: Does the tangent line to ...
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1answer
81 views

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
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2answers
69 views

Differentiating the Taylor expansion of $e^x$

It is well known that a) $\frac{d}{dx}\exp x = \exp x$ and b) $\exp x = \sum\limits_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + ...$. Therefore, it should be possible to ...
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24 views

Taylor series expansions of f(ax, y + dy)

I'm required to do a Taylor series expansion of $f(ax, y+ \delta y)$ where $a$ is a constant and $\delta y$ is an increment of $y$. How would it be done? This is probably a special case of the ...