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.

learn more… | top users | synonyms (1)

1
vote
2answers
289 views

Taylor Series Approximation for degree k Taylor polynomial?

Let $T_k(x)$ be the degree $k$ Taylor polynomial of the function $f(x)=\sin(x)$ at $a=0$. Suppose you approximate $f(x)$ by $T_k(x)$. If $|x|\le 1$, how many terms are needed (that is, what is $k$) ...
1
vote
4answers
81 views

Differentiating second order term of Taylor polynomial (multivariable)

I am trying to derive Newton step in an iterative optimization. I know the step is: $$\Delta x=-H^{-1}g$$ where H is Hessian and $g$ is gradient of a vector function $f(x)$ at $x$. I also know the ...
0
votes
2answers
49 views

Version of Taylor: $F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h \right \rangle.$

My teacher claimed without proof that Taylor's theorem with remainder implied that for a suitable function $F: \mathbb{R}^n \to \mathbb{R}$, $$F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h ...
0
votes
1answer
103 views

Compute the 10th derivative

$f(x) = (\cos(5x^2) - 1 )/ x^2 $ at $x = 0$ We were given the hint to use the MacLaurin series for f(x). I get how to do it if it was just $\cos(5x^2)$ but what would I do with the other values in ...
1
vote
1answer
775 views

Find the Maclaurin series of the function f(x) = (7 x^2) sin (2 x)

Find the Maclaurin series of the function $f(x) = (7 x^2) sin (2 x)$ $(f(x) = \sum_{n=0}^{\infty} c_n x^n) $ That is what is given on the question, we have to fill in 5 blanks $c_3$ to $c_7$ The ...
6
votes
2answers
144 views

Gamma Type Integral

I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ...
0
votes
1answer
46 views

using Taylor's formula in a proof

Prove that $1+\frac{1}{n} < e$ for all $n$ in the natural numbers. How does this connect to Taylor's formula? I know that $e^x > 1+x$ for $x>0$, but then where does Taylor's formula come in ...
1
vote
0answers
39 views

Taylor series convergence

$$f(z)=\int^z_0 \frac{\zeta-\sin(\zeta)}{\zeta^2+4} \, d\zeta$$ I am supposed to find the convergence radius of its Taylor series at point $a=2$. I can find the radius in simple cases by finding ...
4
votes
3answers
114 views

What is the relationship between saying “a Taylor series converges for all $x$” and “a Taylor series converges to a function, f(x)”

Given the following Taylor series: $1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}- \dots$ We know that: It converges for all of $x$ It converges to the function $\cos x$ The ...
2
votes
0answers
55 views

Approximating polynomial of higher degree

Suppose that we approximate a function $f(x)$ for $x$ near $0$ by a polynomial of degree $n$: $$f(x)\approx P_n(x)=C_0+C_1x+C_xx^2 + \dots + C_{n-1}x^{n-1} +C_nx^n$$ We need to find the values ...
0
votes
1answer
76 views

Finding a generating function for a pattern

I was working on this projecteuler.com problem, and I was very interested by the premise. Essentially, given n terms, find an ...
2
votes
1answer
43 views

Stuck with Taylor expansion of $f(x+x')$

I know that the Taylor series of $f(x)$ around $a$ is given by: $$f(x)=f(a)+f'(a)(x-a)+f''(a)\frac{(x-a)^2}{2}+\dots=\sum_{n=0}^\infty \frac{f^{(n)}(a) }{n!} (x-a)^n$$ In my textbook I see the ...
3
votes
2answers
50 views

Maclaurin series of: $ f(x) = {x + 5\over1-x^2}$.

I'm trying to get the Maclauren series of: $ f(x) = {x + 5\over1-x^2}$. I am sure there is some trick here, the result according to Mathematica is: $5 + x + 5x^2 + x^3 + 5x^4 + x^5 + 5x^6 + \ ...$ ...
1
vote
1answer
15 views

finding sequence for e converging at some speed

I want to find an infinite sequence that conerges to e so that the kth term of the sequence is less than 10^-k away from e. Obviously, I've considered the Taylor series, but asymptotic bounds on the ...
3
votes
0answers
46 views

Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
0
votes
1answer
84 views

Show the Newton method converges to 0 quadratically?

Using taylor series, show that if $x_n$ converges to a root, $f(x_n)$ usually converges to 0 quadratically. I reached a point I think I need to show that $\lim_{x\to \infty} ...
7
votes
2answers
133 views

What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
0
votes
1answer
136 views

Taylor series about different points implies different interval of convergence?

I'm considering the taylor series of functions whose radius of convergence is non-infinite about different points, and I'm not sure if I'm interpreting this correctly. Suppose, for concreteness, you ...
4
votes
2answers
184 views

Taylor polynomial about the origin

Find the 3rd degree Taylor polynomial about the origin of $$f(x,y)=\sin (x)\ln(1+y)$$ So I used this formula to calculate it ...
0
votes
1answer
36 views

Confusion related to Taylor series approximation

I found this Taylor series approximation given by $f(x_{\alpha}) = f(x) + \nabla f(x)'(x_{\alpha}-x) + o(||x_{\alpha}-x||)$. I didn't get how this $o(||x_{\alpha}-x||)$ term came from. Can anyone ...
2
votes
1answer
156 views

Explaining and using the $N$-term Taylor series for $\sin x$

So I'm given the Taylor Series expansion of the sine function and I've been asked to prove it (Done) and then construct the following by my lecturer: Explain why the Taylor series containing $N$ ...
4
votes
1answer
138 views

Maclaurin series for $e^z /\cos z$.

I want to find the Maclaurin series for the function $$f(z)=\frac{e^z}{\cos z}.$$ Right away I can tell that the radius of convergence will be $\pi/2$, since it's the distance to the nearest ...
0
votes
1answer
223 views

What is the difference between Taylor series and Laurent series?

Can someone intuitively describe what is the difference between Taylor series and Laurent series? Also, what is the most general formula for both?
1
vote
3answers
395 views

Taylor expansion and error?

This came up in a part of the proof. $-\log(1-x)$ is $x$ and then want to calculate the error of this. The idea is that taylor series of $-\log(1-x)=x+\dfrac{x^2}{2}+\dfrac{x^3}{3}+...$ We have ...
0
votes
2answers
41 views

Taylor evaluation in a product solving a limit

I have the following function, which I am supposed to evaluate: $\lim_{x \to 0}{\frac{(e^{-x^2}-1)\sin x}{x \ln (1+x^2)}}$ My though is to replace sin x by its Maclaurin polynomial, as such: ...
2
votes
2answers
923 views

Natural Logarithm Taylor Series Expansion

f(x)=x$^3$ln(1+2x) Write the first four non-zero terms of the Taylor Series for the above function with x centered at a=0. Using this model: ln(1+x) = Σ$\frac{(-1)^{k}(x)^{k}}{k}$ I get the ...
1
vote
1answer
233 views

How many iterations of Taylor series for n correct decimal digits

I'm using Taylor series to estimate trigonometric functions. So I need to know exactly how many iterations of Taylor series (say for sine) are needed for n decimal digits precision? (I'm writing a ...
2
votes
1answer
168 views

Complex Analysis - Taylor Series

The question reads as follows: Let the function $f$ be given by $f(z)=\frac{z}{2}+\frac{z}{e^z-1}$ if $z\neq0$ and $f(z)=1$ if $z=0$. Show that $f$ is analytic at $z=0$ and that $f(z)=f(-z)$. ...
2
votes
1answer
55 views

Finding Taylor approximation for $x^4e^{-x^3}$

I'm trying to find Taylor approximation for the function: $$x^4e^{-x^3}$$ I started taking the first, second, third, etc. derivatives but the expression for it seems to explode with terms. I was just ...
0
votes
1answer
56 views

Does the Taylor Polynomial approximation work for non-convergent functions?

The approximation of $f(x)$ by $P_{n}(x)$ at $c$ has an error of $R_{n}(x) = \frac{f^{n+1}(z) (x-c)^{n+1}}{(n+1)!}$. Does this work for any $(n+1)$ differentiable function even if it doesn't have a ...
0
votes
2answers
385 views

Finding Taylor's expansion for $f(x) = \sqrt{1 + x} -\sqrt{ 1 - x}$

I know I have to find the derivatives of $ f(x) $ (i.e. $f'(x)$ ..) but I'm confused about what to do afterwards .
3
votes
2answers
192 views

Is there a closed form expression for the Taylor series of exp((f(z))?

Given a holomorphic function $f(z) = \sum_{k=0}^\infty f_k z^k/k!$, is there a readable formula for the Taylor series of $\exp(f(z))$? Using the chain and product rules, one can obtain $$\partial_z ...
0
votes
1answer
112 views

Nonlinear initial-boundary value problems using Taylor expansion of parameter

Let $u(x,t; \epsilon)$ satisfy the nonlinear initial boundary value problem $$ u_{tt} = (u_{x} + u_{x}^3)_{x} + u_{xxt}, \space 0 \lt x \lt 1 $$ $$ u(0,t) = 0 \\ u(1,t) = 0 \\ u(x,0) = \epsilon f(x) ...
2
votes
1answer
221 views

Intuition regarding Taylor series for $\frac{e^z}{1-3z}$.

The question asks me to find the Taylor series for $$f(z)=\frac{e^z}{1-3z}.$$ The radius of convergence is $|z|<1/3$ and I know the expansions for $e^z$ and $1/(1-3z)$ are \begin{align} e^z ...
0
votes
0answers
92 views

A function with an identically zero Maclaurin series

We just recently went over Taylor's theorem in my analysis class, and my professor gave the function $$ f(x) = \begin{cases} e^{-1/x^2}, & \text{if }x \not= 0, \\ 0, & \text{if }x\text{ = ...
1
vote
1answer
64 views

Rotated functions and taylor series

If one rotates a function such as the sine function about the origin, is there a general method to find the taylor series for the rotated function? Assuming of course that the rotated function is ...
0
votes
1answer
205 views

Algorithm for estimating $\beta$ using a Taylor series expansion

I am working on the following question for a mathematical economics class. Consider an econometric model: $$y_t=f(x_t,\beta) + e_t,t=1,...,T$$ where $\{ e_t \}$ is a sequence of mean-zero ...
1
vote
2answers
229 views

Find the two variable Maclaurin series for $f(x,y) = e^{x+y}$

I'm shaky with Taylor/Maclaurin series, and I've been over and over my book and notes and still feel like I'm at square one...
4
votes
0answers
43 views

Am I missing anything when doing this taylor expansion?

I'll make my question short, I am encountering a error when doing expansion. I am expanding $f(x)=2x^3+4x+1$ and after the expansion, things don't match. Here's what I'm doing. Let $a=5$ ...
2
votes
1answer
71 views

Proofs using Taylor Series Expansion

I would just like some help on the theory of maths. If a question asks for proof of a function using the Taylor Series Expansion, can you use the Maclaurin Series? Is using the Maclaurin Series ...
2
votes
2answers
2k views

Taylor Series Expansion for $\sin^2(\omega t)$

What are the first few terms for the Taylor Series Expansion for $\sin^2(\omega t)$? $(\omega$=$2\pi f$) If you could show some working, that would be helpful
3
votes
1answer
91 views

show that $\lim_{x\to \frac14} \frac{d^{2k-1}}{dx^{2k-1}}\cot(\pi x)=-(2\pi)^{2k-1}2^{2k}(2^{2k}-1) \frac{\left | B_{2k} \right |}{2k} $

I try to prove the relation between Polygamma function and Bernoulli numbers but I faced this problem,is how to show that $$\lim_{x\to \frac14} \frac{d^{2k-1}}{dx^{2k-1}}\cot(\pi ...
0
votes
2answers
53 views

Taylor series problem

I have this equation: 960 - 84.60 * ((1-(1+i)^-12)/i) == 0 I simplify ( 1+i)^-12 with a Taylor series ...
4
votes
1answer
235 views

Why do we use big Oh in taylor series?

In the taylor series for sin(x), we write: $$ \sin{x} = x + \frac{x^3}{6} + \frac{x^5}{120} + O(x^7) $$ Meaning that $\sin{x} = x + \frac{x^3}{6} + \frac{x^5}{120}$ and terms of order $x^7$ and ...
3
votes
4answers
163 views

Taylor series of $\arctan(x+2)$ at $x=\infty$

The simple question is: what is the correct way to calculate the series expansion of $\arctan(x+2)$ at $x=\infty$ without strange (and maybe wrong) tricks? Read further only if you want more details. ...
0
votes
2answers
85 views

What does the taylor series give us ?

To be more clear if we use the taylor series for x=2 it will give us an approximation of f(2) ? And why do we stop "adding" the f'''(a)/3! ...? Is there a rule that tells us when to stop ?
1
vote
1answer
42 views

A basic doubt on derivatives

I have one question regarding differentiation : 1) Why in the definition of Taylor's series it requires the function to be "continuously" differentiable $m$ times in $[a,b]$? The book I am following ...
0
votes
3answers
822 views

Derivation of multivariable Taylor series

I am having trouble grokking why it is, assuming that the function is analytic everywhere (and many other assumptions that I am, no doubt, naively assuming), that this is true: ...
1
vote
1answer
42 views

Taylor Series and equation

I have this equation: $$960 - 84.60 \cdot \frac{1-(1+i)^{-12}}{i} = 0$$ I simplify $( 1+i)^{-12}$ with a Taylor series $( 1 + x)^a$. but I obtain $i = 0.087201167$ but the real result should be $i ...
1
vote
1answer
54 views

Taylor expansion of $((H+\epsilon A)^T R^{-1} (H+\epsilon A))^{-1} (H+\epsilon A)^T R^{-1}$

I have seen a kind of contradiction in a paper and I decided to rewrite the equations... Could you please help me to be sure about what I am doing... Let us define $H^\dagger \triangleq ...