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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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
64 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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1answer
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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466 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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52 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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50 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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1answer
70 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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1answer
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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24 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
27 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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35 views

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
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
46 views

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
35 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
64 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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13 views

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
18 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
23 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
48 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
23 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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50 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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1answer
21 views

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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1answer
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
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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
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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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1answer
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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103 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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38 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
31 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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2answers
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) & = & ...
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1answer
47 views

Taylor expansion $\ln(1+x+x^2)$ about $x=0$

Is it applicable to use the taylor expansion of $\ln(1+t)$ here and say $t=x+x^2 $ or do I have to take the derivatives?
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Taylor Expansion of Composition of Functions

I have in my lecture notes that $$f(g(x+h))-f(g(x))\approx f'(g(x))\left(g(x+h)-g(x)\right)+f''(g(x))(g(x+h)-g(x))^2$$ He explained can found via taylor expansion, but I try to expand it and am not ...
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51 views

Determine the Taylor Series for $(1+x)^n$ about $x=0$

Having trouble solving this. I get to expanding to this: $$1^n + n(1^{n-1})\cdot\frac {x!}{1!}+n(n-1)\cdot 1^{n-2} \cdot \frac {x^2}{2!} +n(n-1)(n-2)\cdot 1^{n-3}\cdot \frac {x^3}{3!}\dots$$ Where do ...
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78 views

How do i expand/simplify this quadratic (or quartic?) equation

I'm having trouble doing the following question, was wondering if anyone was able to lend a hand, would be greatly appreciated as i'm not too sure where to start or how to go about this. The ...
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38 views

Taylor Series Formulae

How are the two following forms of the Taylor expansion equivalent? The one I've learnt is $$f(x+h)=f(x)+hf'(x)+\frac{h^2}{2!}f''(x)+...$$ But I've now come across the version $$ ...
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1answer
56 views

Maclaurin Series Approximation of $\sin{x}$

Use first ten terms of the Maclaurin series for $\sin{x}$ to find an approximation to the values of both $\sin{\left(\frac{6\pi}{7}\right)}$ and $\sin{\left(\frac{20\pi}{7}\right)}$? One can say that ...
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33 views

Taylor polynomial manipulation

Find $\sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{k}$ This is in a section in my book on Taylor polynomials/Taylor series so I assume we have to find some way to manipulate Taylor polynomials to get this. ...
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5answers
153 views

Estimate $\int^1_0 e^{-x^2}\, dx$

Estimate $\int^1_0 e^{-x^2}\, dx$ This is in a section on Taylor series so I would assume that is how it should be solved. I started by using the Taylor series formula for $e^x$ replacing $x$ with ...
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1answer
44 views

Using Taylor series find derivatives of arctan(x)

Using Taylor series for $arctan(x)$, find $f^{(5)}(0)$ and $f^{(6)}(0)$ for $f(x)=arctan(x)$ I figure for this problem I compare the general Taylor series formula to the Taylor series for $arctan(x)$ ...
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1answer
62 views

Are there only a few 'universally convergent' Taylor Series?

The taylor series for $sin(x)$, centered at any point, converges for all x. The taylor series for $e^{x}$ and $cos(x)$ do as well. Thus, taking an algebraic function of these (without division) ...
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145 views

Find the value of $a$, $b$ and $c$ for the given limit.

Question - Find the values of $a$, $b$ and $c$ so that $$ \lim_{x\to 0} \cfrac{ae^x - b\cos x +c e^{-x} }{x\sin x} = 2 $$ This is what I've tried yet : For $ x\to 0 $ the numerator must also ...
3
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1answer
45 views

Can this expression of e be simplified?

Using the maclaurin expansions of coshx and sinhx I came up with $e^x = \sum_{n=0}^\infty$${x^{2n}(2n+1+x)}\over {(2n+1)!}$ Plugging in $x=1$ I got: $$e = \sum_{n=0}^\infty {2(n+1)\over (2n+1)!}$$ ...