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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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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66 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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30 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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71 views

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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43 views

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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2answers
31 views

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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40 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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76 views

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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29 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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15 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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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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62 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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4k views

Prove Taylor expansion with mean value theorem

On http://vapour-trail.blogspot.com/2006/03/brief-explanation-of-taylor-series-via.html one can find an hint at how to derive Taylor expansions from the mean value theorem. The process goes as ...
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458 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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3answers
33 views

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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63 views

Taylor's theorem in C

I've got simple code for Taylor's Theorem for cosh() function. I'm trying to catch a mistake - the result is about half the real answer. How can I do it correctly? ...
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48 views

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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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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49 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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68 views

Proof of lagrange inversion of taylor series

is there a proof for the lagrange inversion of taylor series? The formula is given in http://en.wikipedia.org/wiki/Lagrange_inversion_theorem#Theorem_statement The proof cannot be found in the ...
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24 views

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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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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23 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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1answer
26 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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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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2k views

Expected value of a function of a random variable: help!

I am trying to show the following: \begin{equation*} E[e^{-\gamma W}]=e^{-\gamma(E[W]-\frac{\gamma}{2}Var [W])} \end{equation*} but I really can't remember what I am supposed to do to get from the ...
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1answer
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Confusion regarding term in taylor series expansion for dy/dx=f(h)

I start by considering a differential equation $\frac{dy}{dt}=f(y), y(t_0)=y_0$ and using a step size of $\frac{h}{n}$ where h is any arbitrary constant. The 1st step in Euler method will be ...
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1answer
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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
62 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 ...
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Solution to Equation involving Volatility

The following question will have little context, though, it is not relevant. To summarise though, I am trying to find solutions $u$ and $d$ to the following equation given that $d = \frac{1}{u}$: ...
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1answer
17 views

Taylor expansion of second order

I have to find the Taylor expansion of second order of the following functions with center the given point $(x_0, y_0)$. $f(x, y)=(x+y)^2, x_0=0, y_0=0$ $f(x, y)=e^{-x^2-y^2}\cos (xy), x_0=0, ...
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1answer
22 views

Determine Taylor's formula for multivariable equation

Determine Taylor’s formula for the function $f(x, y) = \ln(x+ 2y)$ at the point (1, 0) with remainder term of order three (i.e., the remainder term contains the third total differential). I haven't ...
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1answer
46 views

Maclaurin series for a function

Provided I have the function \begin{equation*} f(x)=(1+x)^{1/x}, \end{equation*} and I want to calculate a 3rd order Maclaurin series, how can that be done without taking direct derivatives (as ...
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1answer
22 views

Power series expansion requirements

Hello stackexchange folks :) I have a question regarding the assumptions made right before you choose to expand or approximate a function by a power series. Specifically I have the function: ...
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49 views

Prove $|f'(\frac{1}{2})|\leq \frac{1}{4}$ [duplicate]

Let $f:[0,1]\to \mathbb{R}$ be a function whose second derivative $f'(x)$ is continuous on $[0,1]$. Suppose $f(0)=f(1)=0$ and that $|f''(x)|\leq 1$ for all $x\in [0,1]$. Prove that ...
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How can I show the remainder of this Taylor polynomial $R(h)/h^2$ goes to $0$ as $h$ goes to $0$?

Given the function $f(x, y) = \frac 1{2 - x - y^2}$ I found that the second-degree Taylor polynomial is $$P(x, y) =\frac12 + \frac{x}4 + \frac{x^2}{4} + \frac{y^2}2.$$ How can I show the remainder ...
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77 views

Calculate the Taylor series of $f(x) =\ln( 1 -x +x^2) $ and the domain of convergence

I just stuck at the following exercise: Show that the function f has a Taylor series and calculate it, with $x_0 = 0$. $$ f(x) = \ln{(1-x+x^2)}$$ Because I already know the Taylor series from ...
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A Taylor expansion of $F(x+f(x))$ when $f(x)$ is small

Let's suppose I have a function $F(x)$ and an invertible function $f(x)$. Denote $y=f(x)$ and $u=x+y$. Does the following Taylor expansion (up to two terms) centered at $y=0$ make sense? $$ ...
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1answer
19 views

Order $n^{r-1}$ approximation of product given order $(\frac{1}{n^2})$ approximation of terms

I have that $|a_n - (1+\frac{r}n)| \leq \frac c{n^2}$, for $c$ a constant, and am attempting to show that there exist constants $C < \infty$ and $K > 0$ such that the product $b_n = ...
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1answer
17 views

Determing taylor series from other series

Consider $\cos(x)$ and $\cos(3x^2)$. How to determine the latter's Taylor series from the formers at $a = 0$? I'd write $$\cos{x} = \sum_0^\infty (-1)^n\frac{x^{2n}}{(2n)!}$$ Now, I could just ...
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22 views

Does $o(|x-a|^n)$ approximation by a polynomial imply existence of derivatives?

While reviewing the topic of Taylor expansion, I've noticed that while in all statements about the $n$th order Taylor polynomial of $f:\mathbb R \to \mathbb R $, it's always assumed that $f\in C^n$, ...
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98 views

Compute limit using Taylor's expansion

Using Taylor’s expansion, prove that the following limit exists and compute it. $$\lim_{x \to 0}\left(\frac {x^2}{\frac {1}{1-x} - e^x}\right)$$ In this if I am using the taylor series expansion ...
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2answers
33 views

Calculating Lagrange error of a Taylor polynomial approximation

So I am slightly confused when it comes to finding the error of a Taylor series approximation. I know the equation is : $ E_n(x)=\frac M {(n+1)!}(x-a)^{n-1} $ where a is the point that it is ...
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37 views

Find the integer $'n'$ for which the given limit is a finite non-zero number.

Find the integer $'n'$ for which the given limit is a finite non-zero number. $$\lim_{x\to 0} \cfrac{\cos^2 x -\cos x -e^x \cos x + e^x - \frac{x^3}{2}}{x^n}$$ I'm almost blind regarding ...
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1answer
29 views

Mc Lauren - Runge Kutta relationship [duplicate]

my question is quite easy(I think). I understand how to apply 4th order Runge Kutta and understand the principle of taylor series (the Mc lauren to be precise) . But I cannot fully understand how the ...
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57 views

Easy way to remember Taylor Series for log(1+x)?

Assuming $|x|<1$, if one can easily remember that $$ \dfrac{1}{1-x}=\sum_{n=1}^{\infty}x^{n} $$ then it's easy to mentally derive the following \begin{eqnarray*} \mbox{log}(1-x) & = & ...