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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4
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1answer
68 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 ...
4
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3answers
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 ...
4
votes
1answer
81 views

Fundamental proof of Taylor's theorem using little-o notations

Is there a fundamental proof of Taylor's theorem using little-o notation? I assume $f:E\rightarrow F$ as a mapping between Banach spaces and write $(h^i)$ for $(h,\ldots,h)$ ($i$ times iterated). ...
4
votes
2answers
82 views

Expansion of $f(x)=\frac {\ln(1+\sin ax)-x(1+\arctan x)^{1/x}}{1-\cos x} $ for $x$ near $0$

How to find finite expansion of $$f(x)=\frac {\ln(1+\sin(ax))-x(1+\arctan x)^{1/x}}{1-\cos x} $$ to order $2$ and neighborhood $0$. My Dr. didn't expand them all to order $2$, please I have problem ...
4
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1answer
97 views

Remainder in the multivariate Taylor expansion

For the function $f:\mathbb{R}\to\mathbb{R}$, I can write the Taylor expansion $$f(x+h) = f(x) + f'(x)h + \frac{1}{2!}f''(x)h^2 + O(h^3)$$ where the remainder is $o(h^2)$ as well. I'm confused with ...
4
votes
1answer
203 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 ...
4
votes
1answer
473 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 ...
4
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1answer
212 views

Question about a solution to a problem involving Taylor's theorem and local minimum

I've been studying "Berkeley Problems in Mathematics, Souza, Silva" and I came across this problem: Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Assume that $f(x)$ has a ...
4
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2answers
179 views

How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?

I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
4
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1answer
96 views

Let $f(x)=e^{-1/x}$ for $x>0$ and $f(x)=0$ for $x\leq 0$. Prove that $f^{(n)}(0)=0$ for all $n\in \Bbb N$.

Let $f(x)=e^{-1/x}$ for $x>0$ and $f(x)=0$ for $x\leq 0$. Prove that $f^{(n)}(0)=0$ for all $n\in \Bbb N$. I'm reading the solution, and I understand how to prove that all derivates must be of the ...
4
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0answers
47 views

Limit and Taylor Series when non-differentiable

I am stuck on a problem similar to this one. Define $$f(\theta,y)=\frac{g(\theta y)}{\int_0^1 g(\theta x)dx}$$ with $g(0)=0$ and $\lim_{t \to 0}g'(t)=+\infty$. I am interested in $\lim_{\theta \to ...
4
votes
2answers
58 views

What happens when I convert a Taylor series into an integral?

Suppose we have the Taylor series of an analytic function: $$f(x) = \sum_{k=0}^\infty \frac{1}{k!} a_k x^k$$ Then I decide to (kind of) turn it into an integral: $$g(x) = \int_0^\infty ...
4
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0answers
78 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
4
votes
1answer
50 views

(Though?)Expression Rearranging

I have the following expression $ 2x+3x^2+e^{5x+x^2}=7 $ which I need rearranged in a form of the type $Ye^Y=Z$ with Y a function of x and Z some constant. I have tried the substitution $y=5x+x^2$, ...
4
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0answers
206 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
4
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0answers
250 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} ...
4
votes
2answers
95 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 ...
4
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0answers
226 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
4
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0answers
95 views

Inverse of a power series

I want to find the inverse function of the power series, $$ f(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!} $$ The only think I can think of that could possibly help is that $$ ...
4
votes
1answer
90 views

How to find the series $\sum_{n=1}^{\infty}\frac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$

Find this sum $$\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n}.\qquad (-1\le x\le 1)$$ My idea: let $$f(x)=\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$$ then we have ...
4
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0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
4
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0answers
48 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 $a=5$ ...
4
votes
2answers
445 views

Range of the sine function

It is obvious from the definition of $f(x)=\sin(x)$ using the unit circle of radius $1$ that the range of that function is the set $[-1,1]$. But also there are approaches where the sine is defined ...
4
votes
1answer
166 views

Write down the equation of the tangent plane and compute the Taylor series of the function

Set $f(x,y,z) = x + y + z + x^2 + y^2 + z^2$. Consider the surface $$S = \{f(x,y,z) = 0\} \subset \mathbb{R}^3$$ near the origin $o = (0,0,0) \in S$. Write down the equation of the ...
4
votes
1answer
119 views

A sort of “Taylor expansion” of a power series

I have the following question. Suppose $$f(x):=\sum_{i=0}^{\infty}c_ix^i$$ is a power series that converges for $|x|<1 + \epsilon$, for some $\epsilon >0$, where $x\in\mathbb{C}$. I can then ...
4
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0answers
489 views

Fastest convergence Series which approximates function

The question is the following: Is there any proof that shows that the Taylor series of an analytical function is the series with the fastest convergence to that function? The motivation to this ...
3
votes
2answers
477 views

Exponential function-like Taylor series: what is it?

I have a series $$1+ x+\frac{x^2}{2}+\frac{x^3}{4}+\frac{x^4}{8}...=1+\sum_{n=1}^\infty \frac{x^n}{2^{n-1}}$$ that looks an awful lot like a Taylor series of some kind. If the denominator of the ...
3
votes
5answers
121 views

Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that ...
3
votes
3answers
715 views

Taylor series for different points… how do they look?

I can't understand what it means to do the Taylor series at the point $a$. The best way would be showing me how it looks for different $a$ on a graph. Do I find those graphs on the Internet?
3
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4answers
4k views

Maclaurin Series of $1/(1-x)$ derived from maclaurin series of $(1+x)^n$

Is there a way to derive the Maclaurin series for $\frac{1}{(1-x)}$ after finding the Maclaurin series for $(1+x)^n$ which is $\displaystyle\sum\limits_{k=0}^\infty \frac{f^k(0)}{k!}*x^k$. From ...
3
votes
5answers
102 views

When $a\to \infty$, $\sqrt{a^2+4}$ behaves as $a+\frac{2}{a}$?

What does it mean that $\,\,f(a)=\sqrt{a^2+4}\,\,$ behaves as $\,a+\dfrac{2}{a},\,$ as $a\to \infty$? How can this be justified? Thanks.
3
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2answers
243 views

Taylor Series of $\tan x$

I found a nice general formula for the Taylor series of $\tan x$: $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$ where $B_n$ are the Bernoulli ...
3
votes
2answers
103 views

Show that $f(x)=e^x$

In this case $f(x)=1+x+x^2/2!+x^3/3!+x^4/4! + ... = \sum_{n=0}^\infty \frac{x^n}{n!}$. I understand it conceptually in terms of the Taylor series, but I have no idea how to prove it rigorously.
3
votes
4answers
78 views

Taylor series of $(1+x)\ln(1+x)$ in $x=0$

How to determine the Taylor series of $(1+x)\ln(1+x)$ in $x=0$? My idea is finding the second derivative of the expression, which is $\frac{1}{1+x}$. The Taylor series of this expression is ...
3
votes
2answers
61 views

Finding the sum of a Taylor expansion

I want to find the following sum: $$ \sum\limits_{k=0}^\infty (-1)^k \frac{(\ln{4})^k}{k!} $$ I decided to substitute $x = \ln{4}$: $$ \sum\limits_{k=0}^\infty (-1)^k \frac{x^k}{k!} $$ The first ...
3
votes
2answers
624 views

Why is Taylor series expansion for $1/(1-x)$ valid only for $x \in (-1, 1)$?

After finding an expansion of $$ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots$$ a quick test of various values for $x$ reveals that this expansion is not valid for $\forall x \in \mathbb{R}-\{1\}$. ...
3
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2answers
470 views

Odd powers in the Taylor series expansion of $\operatorname{Log}(1+e^z)$.

I noticed a question while reviewing Taylor series expansions that has been bugging me. The question is: How many nonzero terms with odd exponents does the Taylor series $\operatorname{Log} (1 + e^z)$ ...
3
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3answers
118 views

definition of the constant $e$

To my knowledge there are two possible ways to define $e^x$ $$e^x = \sum_{i=0}^{\infty}\frac{x^i}{i!}$$ $$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n$$ So my question is: Why does… ...
3
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2answers
106 views

Question involving approximation, taylor series and proving

Question: Consider the approximation $$\ln(2)\approx 2\left ( \frac{1}{3}+\frac{1}{3\times 3^{3}}+\frac{1}{5\times 3^{5}} \right )$$ Prove that the error in this approximation is less than ...
3
votes
2answers
146 views

What is the 35th derivative of $f(x) = e^{x^{10}} $at $x = 0$?

I had this question on a quiz and I answered 35!x^5/4! What is the 35th derivative of $f(x) = e^{x^{10}} $at $x = 0$? Use a suitable Taylor Polynomial for $e^x$ at $x = 0$. Express the answer in ...
3
votes
2answers
76 views

Finding terms of a Taylor series where $f(x)$ is a function with a power

I've been stuck with this Taylor series problem for a while now. We have that $$ f(x) = (1 + x^2)^{-2/3} $$ and it's centered at $0$. So what I thought of doing was the $$ \frac{f^{n}(a)(x - ...
3
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3answers
324 views

Find an accurate value of $f(x)=\sqrt{4x^2+x}-2x$ for large values of x. Calculate $\lim_{x\to\infty}f(x)$

My works: $x^2$ can be very large if x is large, thus the function has lose-of-significance error and we need to reformulate it. $$ ...
3
votes
2answers
5k views

What is the Maclaurin series expansion for $\sqrt{x}$?

The derivative of $\sqrt{x}$ doesn't have a defined value at x = 0. How then do I find its maclaurin series expansion? Or can it only be approximated with a Taylor series at some value x != 0?
3
votes
2answers
87 views

Show $ \lim_{n \to \infty} \frac{\ f(y_n)-f(x_n)}{ y_n-x_n} = f'(x_0)$ using Taylor

Let $f:[a, b]\to R$ differentiable at $a<x_0<b$. Using taylor series show that if $x_n \to x_0^-$ and $y_n \to x_0^+$ then $$ \lim_{n \to \infty} \frac{\ f(y_n)-f(x_n)}{ y_n-x_n} = f'(x_0)$$ ...
3
votes
3answers
42 views

Calculating using Taylor's series with Remainder: $ \lim \limits_{x \to 1} \frac{\ln x}{x^2+x-2} $

How to Calculate with Taylor's series with Remainder: $$ \lim \limits_{x \to 1} \frac{\ln x}{x^2+x-2} $$ without using L'Hopital's Rule? Here is what I reached: $$\lim \limits_{x \to 1} \frac{(x-1) ...
3
votes
3answers
238 views

Limit of $\dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ as $x \rightarrow 0$

Find $\lim_{x \to 0} \dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ I came across this limit a long time ago and could easily obtain a straightforward solution ...
3
votes
2answers
73 views

Coincidence of $x-\frac{x^3}{6}$ and $\sin x$ in an interval

Plotting $f(x)=x-\frac{x^3}{6}$ and $g(x)=\sin x$, one can see that these two function are coincide in an interval $I\subset(-\frac{\pi}{2},\frac{\pi}{2})$. On the other hand, Taylor series for $\sin ...
3
votes
4answers
241 views

Trigonometric Coincidence

I Know that using Taylor Series, the formula of $\sin x$ is $$x-x^3/3!+x^5/5!-x^7/7!\cdots,$$ and the unit of $x$ is radian (where $\pi/2$ is right angle). However, the ratio of the circumference ...
3
votes
2answers
94 views

How can we determine the number of terms which we have to take in a series to get a particular accurate?

As I remember , two days ago , there was a question ( here ) asks for calculating this limit $\displaystyle \lim \limits_{x\rightarrow \infty } \frac{x^3}{e^x}$ and the question was answered . of ...
3
votes
3answers
777 views

Finding the Taylor series expansion of $f(z)=\frac{e^{z}-1}{z}$ around $0$

Find the Taylor series expansion of $f(z)=\displaystyle\frac{e^{z}-1}{z}$ around $0$. I have no idea where to start.