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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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 ...
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The Taylor series of the squared logarithm

Prove that the Taylor series about the origin of the function $[\log(1-z)]^2$ is given by $$\sum_{n=1}^{\infty} \frac{2H_{n}}{n+1} z^{n+1}$$ where $$H_{n} = \sum_{j=1}^{n}\frac{1}{j}$$ is ...
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37 views

What is the Taylor series of $e^x$ centred at $3$?

$$ \sum_{k=0}^n \frac{e^3}{n!}(x-3)^n $$ This is my answer - is it correct?
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1answer
17 views

Taylor series at $a = 0$

If we are given a power series with $a = 0$ and it converges to $f$ in some interval around $a$, then the power series is the taylor series to $f$. But what is the taylor polynomial of $f$ to some ...
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Proof of a step of a lemma on the asymptotics of maximum likelihood where a Taylor expansion is used. (crosspost from crossvalidated).

I have asked this question on crossvalidated here and I am still unsure on the answer. I attempt a cross-post (most of the times this proves very useful). I copy the question below: I am trying to ...
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1answer
25 views

Approximation formula for third derivative, is my approach right?

Derive by using Taylor approximation up to 4th degree (in $h$) of $f$ in $x_0 \pm h$, $x_0\pm 2h$ at $x_0$, an formula for approximation of $f'''(x_0)$ with an error term of order $h^2$. Could ...
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42 views

The power series $\sum_{n\geq 1} \frac{x^{n}}{n(2n-1)}$ with $2$nd Taylor polynomial and Taylor series.

Consider the series $$\sum_{n\geq 1} \frac{x^{n}}{ n(2n-1)}\tag{1}$$ and the sum is denoted by $f(x)$. i) I want to determine the second Taylor polynomial for $f(x)$ about $x=0$. ii) I want to ...
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9 views

Taylor series for a multivariable function

We know the following: $a \approx 1 + x\sqrt{dt}$ $V_1 = V(aS, t + dt ) $ The textbook claims you can (using Taylor's Theorem), expand the bottom-most equation like this: $V_1 \approx V + ...
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3answers
232 views

Rewrite trigonometric expression to be be numerically “stable”

Is it possible to write the following function: $$ f(x) = \begin{cases} \frac{x-\sin x}{1- \cos x}& x\neq 0\\ 0 & x=0 \end{cases} $$ as a composition of elementary functions (including ...
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Find a formula for the binomial coefficients of the Macluarin series for $\frac{1}{(1+x)^{1/2}}$

The Maclaurin series for $\frac{1}{(1+x)^{1/2}}$ is \begin{equation*} 1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\frac{35x^4}{128}...~. \end{equation*} I can't figure much out other than it ...
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How to find the largest positive real value r for Maclauren polynomials where error is set

I need some tips on answering this question: The function is $\sin^2 (r)$ Using Taylor's Theorem, determine the largest positive real value r for which we can guarantee that the Maclauren ...
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2answers
29 views

Taylor expansion of a complex function

Trying to find Taylor series of $$\frac{z^2}{(1+z)^2}$$ I write it in the form $1- \frac{2}{1+z} + \frac{1}{(1+z)^2}$ and I can find Taylor expansion for each factor, is there another method without ...
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40 views

Taylor series of $1\over z^2$

How to find the Taylor series of $1\over z^2$ near $2$ ( in the power of $z-2$) I have tried to write it in the form: $1\over ((z-2)^2+4z-4)$ But I reached nothing, any help please
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28 views

Taylor series in complex analysis [on hold]

I am working on finding the Taylor series of $$\frac1{az+b}$$ in powers of $z.$ How to start with it Any help in details...
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Theoretical Question regarding Taylor Expansion

I received the following question during a Calculus $2.0$ course in my university. I am not a native speaker, so please excuse my English. The question is as follows: Let $f$ be a function with ...
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43 views

Laurent series for $\frac{2}{(z)(z-1)(z-2)}$

! So I think I am getting the hang of Laurent Series, but having a bit of trouble with one of the fractions for part a). So I split this up in to partial fractions: $\frac{1}{z} - ...
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1answer
34 views

Explanation of taylor series

I understand that for a Taylor series of a function $f(x)$, centered around the point a, the general expression can be written as: $$ \begin{align} &f(x) \\ &= f(a) + f'(a) (x-a) + ...
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1answer
30 views

Find a power series that will converge to F(x)

The question is: Find a power series that will converge to $F(x) = \int_0^x\sin(t^2)\;dt$. I don't really have any idea how to solve this, but I know that I need to create the Maclaurin expansion so ...
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1answer
35 views

Is there an alternative to Taylor expansion of functions with more control over the error distribution?

As a physics student I see the Taylor series being (ab)used very often, mostly for the expansion of $\exp(x)$. The usual line of though is that the error (remainder) term of a high order is likely ...
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A problem of Taylor series [closed]

I need a step-by-step solution to the following problem. Sorry, I have nothing done because I don't know how to approach the problem. Determine $\alpha \in \mathbb{R}$ such that $$ \arctan^2 (2x) - ...
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26 views

Find Taylor-Maclaurine expansion of function

Find Taylor-Maclaurine expansion of function: $f(x)=\sin(x)\cdot \cos(x) \cdot \arctan x^2$ my try: $f(x)=\frac{1}{2}\sin{2x} \cdot \arctan x^2$ and we have $\displaystyle \sin{2x} = \sum _{n=0} ...
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Solve $x^2=\cos x$ using Taylor series for cosx

I have the following equation:$x^2=\cos x$ and calculating the Taylor series of $3rd$ degree around $0$ I've got: $x\approx \pm\sqrt{\frac{2}{3}}$ However, now I need to prove that if x is a ...
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Taylor series $\ln(1+e^x)$ about $x=0$

What's the best way to determine this up $x^3$ terms? I thought it would be to take the series for $\ln(1+x)$ and the series for $e^x$ up to $x^3$ and sub the second series into the first. ...
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1answer
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Upper bound for truncated taylor series

A paper claimed the following but I can't figure out why it's true: For all $1/2> \delta > 0$, $k\le n^{1/2-\delta}$, and $j\le k-1$ where $n$, $j$, and $k$ are positive integers, the following ...
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Taylor series in two variables?

how can I calculate the taylor series for a two-dimensional function? Example: \begin{equation*} f(x,y) = Log(1+x+y). \end{equation*} I have $f_x = (1+x+y)^{-1} = f_y$. $f_{xx} = -(1+x+y)^{-2} = ...
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1answer
33 views

Taylor Expansion Derivation

I have a (perhaps silly) question, but I am reviewing Taylor series approximation and looking at this slide. In the third panel, $f(a+h)$ is isolated but I am confused why $O(h^{2})$ does not have a ...
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1answer
32 views

Help Regarding The Taylor Series Remainder Proof Understanding

I'm reading Mathematics: It's Content, Methods, and Meanings and I am in a chapter about Taylor Series. It made sense until I came across the remainder part of the theorem. In order to prove the ...
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Need help with taylor series.

Evaluate the limit $$\lim\limits_{x \to 1} \frac{1-x + \ln x}{1+ \cos πx}$$ The limit im trying to get is $-\frac{1}{π^2}$ as I've solved from l'Hopitals rule. Now I need to solve the limit by ...
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When is Taylor series substitution valid?

Given that the Maclaurin series for $g(x) = \frac{1}{1+x}$ is $1 - x + x^2 - x^3 + x^4 ... $, I'm told that the Maclaurin series for $\frac{1}{1+x^2}$ is $1 - x^2 + x^4 - x^6 ... $, by substituting in ...
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25 views

Write down the Maclaurin series for the function.

(d) Another function is defined on the interval of convergence of the power series in (b) by the formula $\displaystyle{g(x)=\int_0^x \ln(2t+1) dt}$. Write down the Maclaurin series for the ...
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1answer
11 views

cubic approximation with four points(approximating sine function with polynomials)

I was reading the following article https://mixedmath.wordpress.com/2013/11/17/an-intuitive-overview-of-taylor-series/ regarding Taylor Series.When I got to the part 1.3. Cubic approximation I got ...
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44 views

Why the general formula of Taylor series for $ln(x)$ does not work for $n=0$?

I need to find the taylor series for $log(x)$ about $a = 2$, and I have find the following solution, but I don't understand why the general formula does not work for $n = 0$.
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1answer
16 views

Maclaurin series of $y=\ln (\dfrac{1+e^{-x}}{2})$ with $\dfrac{dy}{dx}=\dfrac{e^{-y}}{2}-1$

Maclaurin series of $y=\ln (\dfrac{1+e^{-x}}{2})$ with $\dfrac{dy}{dx}=\dfrac{e^{-y}}{2}-1$ This question requires you to use the given result of $dy/dx$ I've worked out that ...
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Maclaurin Series for $\tan x$ using $\sin x$ and $\cos x$

Now I know the Maclaurin series for $\sin x$ and $\cos x$. Without expanding the series, how can one use the general expression for the two Macluarin series to come up with one for $\tan x$. Dividing ...
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1answer
50 views

Why Does The Taylor Remainder Formula Work?

I've been studying calculus on my own and have come across Taylor series. It is very intuitive until I came across the remainder part of the formula where things got fuzzy. I understand why the ...
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Partial derivatives of the exponential-multivariable

Consider the function $u(x,y)=e^{x^4y^6}$. How do I calculate $\frac{\partial^{35}u}{\partial x^{20} \partial y^{15}}(0,0)$ using Taylor series?
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Proof verification+proposition

Given 2 function $F(p,v)$ and $\frac{dF}{dv}=g(p,v)$ Differentiate F(p,v) with respect to v give $F_pf+F_v$ Formula 1 $$\frac{dF}{dv}=F_p\left(\frac{dp}{dv}\right)+F_v=g\\ ...
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Maclurin Series. (Approximation)

Given that $y=\ln \cos x$, show that the first non-zero terms of Maclurin's series for $y=-\frac{x^2}{2}-\frac{x^4}{12}$. Use this series to find the approximation in terms of $\pi$ for $\ln 2$. My ...
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428 views

Application of Taylor's Theorem in Number Theory

I'm working through Alan Baker's book A Concise Introduction to the Theory of Numbers, and there's an assertion in there that confuses me. Here's the quote: It is easily seen that no polynomial ...
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How do I find the 2nd order Taylor expansion of this function of matrices?

I am looking to form the 2nd order Taylor approximation of the following function of matrices: $$f(W_1,W_2,W_3) = \left\lVert y - g_3(W_3g_2(W_2g_1(W_1x))) \right\rVert_2^2$$ Where: $x \in ...
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Expansion of reciprocal of quadratic

Can I expand $\frac{1}{1-.7B-.3B^2}$ into an infinite series? Where B is the backwards operator in time series. I was thinking $\frac{1}{1-(-.3B)}\frac{1}{1-B}$. Express this as a product of a ...
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Find all real a for which the following is true for all $x > -1$, $ ln(1+x) < x -\frac{ x^2}{2} + ax^3$

Find all real a for which the following is true for all $x >-1 $, ${ln(1+x) }< x -\frac{ x^2}{2} + ax^3$ the question is pretty much as mentioned above. I figured out that the expression is ...
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20 views

Lagrange error bound for a 4th degree polynomial.

A function $g$, which has derivatives for all orders for all real numbers, has a 4th degree Taylor polynomial for $g$ centered at x = 4. The 5th derivative of g satisfies the inequality $g^{5}(x) ≤ ...
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48 views

Linearizing an equation containing both $x$ and $\ln x$

The equation of interest is of the form: $$ k_1 \ln(y/x) = k_2 x $$ And I am wondering how can one linearize this equation for $x.$ Splitting the $\ln$ function would give something along: $$ k_1 \ln ...
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2answers
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confusion regarding 'o' function .

could one explain me the following steps ? my books have written , $$\sec(x) = \frac{1}{1-x^2/2 + o(x^2)} = 1 + x^2/2 + o(x^2)$$ $$\sec^2(x) = \left(1 + x^2/2 + o(x^2)\right)^2 = 1 + x^2 + o(x^2)$$ ...
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Using Maclaurin series for definite integrals

I'm trying to solve this problem but I keep getting the wrong answer. Could anyone check my steps to see if I'm messing something up? Use the Maclaurin series for $e^{-3 x^4}$ to evaluate the ...
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1answer
23 views

Series expansion as a means of 'proving' Simpson's Rule?

I've been working out questions regarding Newton Raphson and Simpson's Rule, whilst they're fairly easy to execute, the latter seems to boggle my mind a little bit more in terms of what the examiner's ...
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1answer
39 views

Complex number, power series

Develop $\sinh z$ in powers of $z-\pi i$ to show that $$\lim_{z\to \pi i}\frac{\sinh z}{z-\pi i}=-1$$ I know that $\sinh z=\sum_{n=1}^\infty \frac{z^{2n-1}}{(2n-1)!}$. Edit: Following the hint ...
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1answer
18 views

Proof using smaller step size and increasing step, Euler method tend to exact solution(solution verification)

Please help to verify is the proof below contain any error. I start by considering a differential equation $\frac{dy}{dt}=f(t)$ and using a step size of $\frac{h}{n}$ where n is consider to be a very ...
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1answer
55 views

Prove equation using taylor series

Given $f(x)$, knowing that $f'(x)$ and $f''(x)$ exist for every $0\leq x\leq1$, and provided I know that $f(0)=f(1)$ and that for each $0\leq x\leq1$, $|f''(x)|\leq A$, how can I prove that ...