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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Taylor theorem remainder term

I'm having trouble applying the formula for the remainder in the Taylor's theorem. From Wikipedia we know that for $f(x)=f(a)+f'(a)(x-a)+…\frac{f^{(n)}(a)}{n!}(x-a)^{n}+R$ the remainder $R$ in the ...
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50 views

L'Hospital's rule vs Taylor series

One classical application of Taylor expansions is to obtain polynomial equivalents of complicated functions and use them to compute limits. For example, with Landau notations, we have ...
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Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
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Find the first three terms of the taylor expansion of $\frac{cos(z)}{1 + z^2}$

The question is: Find the first three non zero terms for the taylor series for $\frac{\cos(z)}{1 + z^2} $ around $z_0 = 0$ What I've done so far is let $f(z) = \frac{\cos(z)}{1 + z^2}$ Then I let ...
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2answers
29 views

Complex functions and Taylor series

Find the Taylor series arround $z_0=0$ write radius of convergence a) $f(z)=\cosh(z)$ b) $f(z)=\log(z+1)$ I don't know how it works with the complex functions. Could you show me the workflow? I ...
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79 views

Is $\cos(\frac{\pi}{3})$ exactly equal to 0.5 or approximately equal to 0.5

We know that $\cos(\frac{\pi}{3})=\frac{1}{2}$, but the expansion for $\cos(x)$ is as follows: $$ \cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\ldots$$ So this would make $$\begin{align} ...
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33 views

Proving an inequality using Taylor's Theorem

I need to show that $ x^{1/3} < \frac{1}{3}x + \frac{2}{3} \forall x \in (0,1)$. I have been given the hint to consider the expression $\frac{1}{3}x - x^{1/3}$, but the Taylor Series centred at ...
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55 views

How do I compute the Taylor Series for $\arctan(x)$?

I've just stumbled upon Taylor Series on Wikipedia and I've been trying to obtain an expansion for $\arctan(x)$, but I can't manage to see a pattern for the $n$th derivative . Can someone come up with ...
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70 views

Taylor development of $\arctan(\cos(x))$ near $0$

How would I find the "Taylor development of $\arctan(cos(x))$ near $0$ at order $5$?" I am translating that from french, so I am not sure how I have to call it it english. By order $5$ I mean that I ...
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27 views

What is the radius of convergence of the derivative of a smooth Taylor series?

On this website I found that the derivative of a Taylor series has the same radius of convergence as the Taylor series itselves. However, there is no reference added, and I seem to be unable to find ...
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26 views

Finding a function with a given Taylor expansion

Is there any function $f(x)$ which has the following Taylor series representation? $$ f(x) = \sum_{k=0}^{\infty}{c_{k} (1 + \frac{2x^{2}}{k})^{-k/2}}. $$ for some coefficients ...
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39 views

Taylor expansion of matrix function

Find the Taylor expansion of the following expression as a function of $C$ around $C_0$ $ GC(I+GC)^{-1}$ in which $G$ and $C$ are matrices of compatible dimensions.
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27 views

Taylor theorem and a $C^{3}$ function with the following property…

A function $f:\mathbb{R} \rightarrow \mathbb{R}$ is $C^3$ with $$f(a+h)=f(a)+f'\left(a+\dfrac{1}{2}h\right)h$$ whenever $a \in \mathbb{R}$ and $h \geq 0$. By applying Taylors Theorem to $f$ and to ...
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Plank's first law expansion

I'm getting a little stuck on this question. The question is: show that for $KT \gg h\omega$, the first law of Planck: $\displaystyle U =\frac{h\omega}{e^{(h\omega/KT)}-1} \approx KT - ...
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45 views

Taylor Polynomials — $\cos(x)$ [closed]

Show that $$\forall x : \cos(x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots+\frac{(-1)^nx^{2n}}{(2n)!}$$ I know that this is true because it is one of the most common Taylor polynomials. ...
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38 views

Wynn-epsilon convergence

How could I use the Wynn-epsilon alghoritm in Matlab to accelerate the convergence of a Maclaurin series? I want to extimate the first derivative of $f(x)$, so $$f'(x)= \sum_{k=0}^\infty ...
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45 views

Taylor series of a division-by-zero equation

I need to calculate taylor series of $(\frac1{t^3}+\frac3{t^2})^{1/3} - \sqrt{(\frac1{t^2}-\frac2{t})}$ at $t = 0$ to calculate limit $(\frac1{t^3}+\frac3{t^2})^{1/3} - ...
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1answer
44 views

Error Estimation Using Taylor's Theorem

I missed the lecture on this and was wondering if someone could explain the steps involved with this problem. I think that what I have to do is evaluate the polynomial up to the second derivative ...
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50 views

Find the Taylor series generated by f at x=a.

$f(x) = \frac 1 {9 - x}, a = 3$. The answer in the book is $$\sum_{n = 0}^{\infty} \frac{(x - 3)^n}{6^{n + 1}}$$but I'm not sure how to get the above.
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1answer
35 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
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121 views

Remainder form of Taylor polynomial at $x_0$: $ \frac{1}{n!}f^{(n)}(x_0 +\theta(x-x_0))(x-x_0)^n$ with $\theta \to \frac{1}{n+1}$ as $ x \to x_0$

If the function $f: \mathbb R \to \mathbb R$ is $n+1$ times differentiable at $x_0$ and $f^{(n+1)}(x_0) \neq 0$, then a form of the remainder in Taylor's Formula is supposedly $$r_n(x_0;x) = ...
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33 views

How is the Harris Corner detector derived from a Taylor Expansion?

It looks more like an assumption about local linearity of the function when the shifts u and v are small. This description of ...
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53 views

Convergence of Taylor Series

My professor made this claim about Taylor Series convergence in my Complex Variables class and I am still not entirely convinced (he said it's explained in the textbook and textbook states, "we will ...
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29 views

Somehow “mirroring” the Taylor-expansion of some $g(x)$

In proceeding with an older discussion of a summation-procedure for divergent series using the matrix of Eulerian numbers I came to a rather general formulation but cannot/do-not-know-how-to make ...
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1answer
78 views

Why does $\arctan(\frac{\tan \theta}{2}) \approx \frac{1}{2 - \theta} - \frac{1}{2 + \theta}$ for small $\theta$?

In answering this question, I needed to show that $\arctan \left( \frac{\tan \theta}{2} \right) \approx \frac{\theta}{2}$ when $\theta$ was small. So, naturally, I computed the first few terms of the ...
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1answer
25 views

Taylor series remainder question

Let $f(x)=\frac{\sin(x)}{x}$ when $x\neq 0$ and $f(x)=1$ when $x=0$. Starting with the Taylor polynomial of degree $2n+1$ for $\sin(x)$ and the estimate for the remainder term, show that ...
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36 views

Analytic Function Derived From Recursive Reverse Taylor Series?

Given the following recursive relation: $a_0 = 1,$ $a_n = a_{n-1}(p-2q)+2(-p)^n$ is there a simple function that has this as its Taylor series, i.e. $f(x) = \sum_{n=0}^{\infty} \frac{a_n}{n!}x^n$ ...
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41 views

Proof with Lagrange Remainder Theorem

I'm trying to prove that $$1+ \frac{x}{3} - \frac{x^2}{9}<(1+x)^{1/3}<1+\frac{x}{3}$$ if $x>0$. Using the Lagrange remainder theorem, I have that $$(1+x)^{1/3}= 1+ \frac{x}{3} - ...
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Taylor Remainder proof for $e^x$

Prove that if $x\leq 0$ then the remainder term $R_{n,0}$ for $e^x$ satisfies $|R_{n,0}|\leq \frac{|x|^{n+1}}{(n+1)!}$. First, $P_{n,0}(x)=1+x+\frac{x^2}{2!}+\cdots+\frac{x^n}{n!}$ with ...
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1answer
41 views

Taylor series of $e^{(x-1)^2}$ about $x=1$

How would we find the Taylor series of $e^{(x-1)^2}$ about $a=1$? I tried finding the answer using the Taylor series of $e^x$ about $a=1$ which I was able to do correctly. When I substituted ...
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70 views

Taylor Series Substitution $e^{x^2-1}$

If I'm using substitution to find a Taylor series about $x=1$ for $e^{x^2-1}$, but I'm given the Maclaurin series for $e^x$, how come the fact that the Taylor series is about $x=1$ doesn't matter when ...
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26 views

Maclaurin series for (cosx-1)/(x^2)

The solution for this is -1/x+x^2/4!-x^2/6!......, but I'm not sure how to derive this Maclaurin series from cos x. The solution just divided each term in the Maclaurin series for cos x by x^2, and ...
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1answer
41 views

Proving Remainder of Taylor Series of 1/(1-x) approaches 0

It is well known that the Taylor (Maclaurin) series of $f(x) = \frac{1}{1-x}$ is $\sum_{n=0}^\infty x^n$ on $(-1,1)$. I am having difficulty proving the equality of these two. The error term is ...
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92 views

Evaluate the limit with Taylor series

How one can evaluate following limit: $\lim_{x\to\infty} x(\frac{1}{e}-(\frac{x}{x+1})^x)$ ? I've found this exercise in the chapter about Taylor series, but I have no idea how to solve it.
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A question about Maclaurin polynomial

Could you please give me some hint how to find 3-th degree Maclaurin polynomial of f(x) given f(0)=1 and for all $0<x<\lambda$ $f'(x)=1+f(x)^{10}$. If $\lim_{x\to0}f(x)=f(0)=1$ then $\lim_{x\to ...
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Using Taylor's Theorem to show $|x-tan(x)|\leq 1/300$ for $0\leq x \leq 1/10$

Using Taylor's Theorem deduce that for $0\leq x \leq 1/10$ $|x-tan(x)|\leq 1/300$ So my attempt; to get the taylors theorem about $x_0=0$ $f(x)=x-tan(x)$ $f'(x)=1-sec^{2}(x)$ ...
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67 views

Approximating the erf function

I was trying to find an approximate solution to the following: $\DeclareMathOperator\erf{erf}$ $$\frac12 \sqrt{\pi} \erf\left (\frac{x-2}{\sqrt{10}}\right) + \frac12 \sqrt{\pi} \erf ...
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236 views

Derivation 9.97 in Jaynes' Probability Theory

In page 298 of Jaynes' Probability Theory: the Logic of Science, equation (9.97), Jaynes says: We expect that, if hypothesis $H$ is true, then $n_k$ will be close to $np_k$, in the sense that the ...
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Is there closed form for $(1-p)(1-p^2)(1-p^3)…$ or its Taylor expansion?

I was considering the following problem: Somebody uses a backup for something (e.g. backups a file) and the backup is equally reliable as original storage. The storage is not perfectly reliable and ...
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Differentiable function made up of arbitrary points.

Hi all, for this question , my attempt so far is; The function $F$ here is considered as a function of $t$ alone; the value of $x$ is regarded as a constant. Of course, if we change the value of $x$ ...
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Find the Taylor Series generated by $\frac1x$ at $x = a$

Can someone help me find the Taylor series for the following equation: $f(x) = \frac1x$ at $a = 10$
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TaylorSeries of complete elliptic integral of the first kind

I want compute $K(k)$ as a Taylor Series; $k\in\mathbb{R}$ and $\vert k \vert < 1$. Can someone help me? $$ K(k):= \int^{\frac{\pi}{2}}_0 \dfrac{dt}{\sqrt{1-k^2 sin^2t}} $$ Results so far: $$ ...
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How to find the full Taylor expansion of the following:

I need to find the full Taylor expansion of $$f(x)=\frac{1+x}{1-2x-x^2}$$ Any help would be appreciated. I'd prefer hints/advice before a full answer is given. I have tried to do partial ...
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1answer
39 views

Proof concerning logs and taylor series

Prove that if $n$ is a positive integer and $|x| \leq \dfrac{1}{2}n$ then $(i)\quad n\log\left(1+\dfrac{x}{n}\right)=x+Q_{n}(x)$ where $(ii)\quad |Q_{n}(x)|\leq\dfrac{|x|^{2}}{n}$ and deduce ...
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49 views

Can a function be approximated by finite number of Taylor expansion terms outside of disk of convergence?

Suppose we have a finite number of terms for Taylor expansion of a conditionally convergent function. For example, $f=\frac1{1-x}$ with expansion $f=\sum_{n=0}^\infty x^n$. This expansion diverges ...
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1answer
47 views

Find the Taylor polynomial of degree 4 for cos(x), for x near 0

I am self studying calculus and I need help solving a Taylor Series problem. 1a) Find the Taylor polynomial of degree 4 for cos(x), for x near 0: I think the answer would be: ...
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1answer
32 views

Maclaurin series and taylor

Im trying to find the first four terms of Maclaurin series of $\space0.15t^2$ and evaluate $$\int_0^1 e^{-0.15}t^2 dt $$ Please this is a revision question. How do i go about it?
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322 views

Euler's identity in matrix form

I assume everyone is familiar with the famous mathematical identity due to L. Euler: $$ e^{i \, \pi} + 1 = 0,$$ where $i^2 = -1$ and $e$ is the base of natural logarithms. I was wondering if this ...
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1answer
35 views

A question on Taylor expansion/approximation

Suppose we are given a continuos function $f(x)$ where $x \in [0,2]$, and the function $f(x)$ is $n$-th-order differentiable, for $n \in \mathbb{N}$ and $n>2$. Besides, we know that these ...
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25 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 ...