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 expansion

Is there an easier way to do a Taylor expansion of $e^{u^2+u}$ than do derivatives or substitute and then use Newton's binomial? For example, expanding until the $4$th term: $$e^{u^2+u}=1+u^2+u+ ...
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1answer
181 views

Thinking through a Taylor error bound for arcsine

In lecture, we went through solving a Taylor error bound for arcsine. I followed most of it except for where it talks about the odds divided by the evens divided by $2n+1$ gaining in accuracy by a ...
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1answer
93 views

Calculating $\arctan(3)$ using Taylor series

I'm trying to get a Taylor series equivalent for $\arctan(3)$, but the standard definition for $\arctan(x)$ is restricted to $|x| \le 1$. How can I get a Taylor series for this expression?
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1answer
2k 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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2answers
85 views

derative of Taylor expansion

I'm reading this part of article about key points localization in image processing, and there is something I don't quite understand, mathematically it's this, $$D(w) = D + {\frac{\partial D}{\partial ...
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1answer
370 views

Numerical analysis Taylor's method question: Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$.

Let $f(x)=\tan^{-1}(x)$ Let $P_n(x)$ be the $n$th Taylor polynomial for $f(x)$ about $x_0=0$ Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$. Is ...
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65 views

Coefficients of series

Suppose that i have a function $f(x)=\sum_{i=0}^{\infty}a_ix^i$ with radius of convergence $r_f>0$ and that i want to write $f$ in a form $f(x)={e^{g(x)}}$ where $e$ is natural logarithm base and ...
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1answer
96 views

Taylor series and its relation to sine

I recently read, one of the most inspiring pieces of literature I've seen, Lockhart's Lament. And now I find myself constantly doing math for fun in my head with imaginary perfect shapes. One such ...
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2answers
152 views

Taylor expansions of $\text{atan}(\tan(x))$ and $\text{asin}(\sin(x))$

Do they actually exist? At least in a form that doesn't degenerate into a mantissa function or into repeated ranges of f(x)=x.
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0answers
77 views

Taylor Polynomial in Multivariable Case

I am doing Problem 9.30 in Rudin's Principles of Mathematical Analysis and have done part (a) and (b) but got stuck on part(c). In part (b) I have finished the proof that: $$f(\mathbf a+\mathbf ...
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1answer
217 views

How to determine the series for $ f(x) = \sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+x}}}} $ around $0$?

In trying to answer a recent MSE-question I came on the partial problem to determine the power series for the function $$ f(x) = \sqrt{1-\sqrt{1+\sqrt{1-\sqrt{1+x}}}} $$ I was not successful ...
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176 views

Taylor expansion with random variables $\frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}$

In Einsteins theory of relativity the kinetic energy of an object is given by the following formula $$E_k = \frac{mc^2}{\sqrt{1-\frac{v}{c}^2}} - mc^2$$ where m is mass of the object at rest v iss ...
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1answer
256 views

trying to understand the incremental form of the Taylor series expansion

original form of Taylor series $f(x)= f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2+ \cdots\\\\$ By making a substitution, we can find something that resembles the incremental form of the ...
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64 views

Geometric Series into Maclaurin Series

Expand 1/(1+x) into Maclaurin Series I found f(0)=1, f'(0)=1, f''(0)=2!, f'''(0)=3! and so on Therefore f^(k)(0)=k! so would the series centered at 0 be equal to x^k ? Just want to check to see if I ...
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3answers
908 views

Taylor and Maclaurin Series for $f(x)=e^x$

I just came from a final exam where in one question I was asked to derive the Taylor Series for $f(x)=e^{2x}$ centered at $x=1$. I came up with the following: ...
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1answer
79 views

Taylor Expansion with Integral Remainder Question

I have the following question at hand and I have to admit that I am not used to integral remainder form of taylor approximation. I am still trying to work around, so a couple of hints would be useful ...
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1answer
271 views

Is exponential function analytic over all complex numbers

In my textbook, I find a text where it says $e^z$ is analytic everywhere (in complex plane). Is it true? If so, what is the proof? I approached using maclaurin series, which gives $e^z= ...
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1answer
104 views

Series Expansion of an Exponential with a Trig Function in the Exponent

Can anyone get a general expression for $$e^{a\cos x}$$ in terms of an infinite sum? I'm having trouble with a general form in terms of $n$ for the coefficients... Alex
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2answers
105 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 ...
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206 views

Why is the Taylor series $\ln(1-2x) = \sum\limits_{k=1}^\infty (-1)^{k+1}\frac{(-2x)^k}{k} $ incorrect?

We know that: $$\ln(1+x) = \sum\limits_{k=1}^\infty (-1)^{k+1}\frac{x^k}{k} $$ Can we replace $x$ by $-2x$ and get: $$\ln(1-2x) = \sum\limits_{k=1}^\infty (-1)^{k+1}\frac{(-2x)^k}{k} $$ this?
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44 views

Find Coefficient in expansion

What will be the coefficient of $x^8$ in the expansion of $x^2\cos x^2$ around $x=0$? I know that it can be done using Leibniz formula for higher order diferentiation of product of two ...
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1answer
67 views

Asymptotic expansion of $\ln\left(\frac{x+a}{x-a}\right)$ in form of $\sum\limits_{n=0}^\infty a_n \left(\frac{1}{x}\right)^n$?

How can I find an expansion for $f(x)=\ln\left(\dfrac{x+a}{x-a}\right)$ in terms of powers of $x$, in the form of: $$f(x)=\sum_{n=0}^\infty a_n \left(\frac{1}{x}\right)^n$$ When I try a Taylor ...
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93 views

$\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}x^k =$

Prove that for $|x|<1$, $\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}(-x^2)^k = ...
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1answer
204 views

Raise a power series to a fractional exponent?

In showing that $\log^\alpha{(1+x)}$ is $O((x)^\alpha)$ at $1$, for $\alpha>0$, one can note that $$\left ( \frac{\log{(1+x)}}{x} \right )^\alpha \overset{x\to 0}{\longrightarrow} \left ( 1\right ...
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2answers
105 views

Error in approximation of nonzero root to $x^2=\sin x$ using Taylor's cubic polynomial.

I have successfully obtained the root's approximation $r=\sqrt{15}-3$ as I'm supposed to as following:$$\begin{align} \displaystyle f(x)=\sin x &= x - \frac{x^3}{6} + E\\ x^2 &\approx x - ...
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0answers
103 views

$\sin(x)$ infinite product [duplicate]

In the equation: $$\sin(x) = x\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)$$ I know that if I do $$0 = x\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right)$$ All the roots are really ...
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1answer
59 views

Applying the Taylor series

For the initial value problem $\dot{y} = f(y), y(t_0) = y_0$, where $f(y)$ is smooth, we look at the discrete evolution $\Psi^t := y_1 = y_0 + h f((1-\Theta)y_0 + \Theta y_1)$, where $\Theta \in ...
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1answer
76 views

What's Taylor expansion of: $f(x)=\frac 1x\ln{(1+2x^2)}$?

What's Taylor development on the next function: $f(x)=\frac 1x\ln{(1+2x^2)}$? Actually this one is the first question I've seen with $ln$, My instincts tell me to try and do derivative in order to ...
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1answer
101 views

Taylor series of a power series.

Consider a power series $f(x)$ around a point $c \neq 0$. Then is it equal to its Taylor series around $0$? The reason I am wondering about this is because if it is true even for some special cases, ...
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1answer
742 views

how do you determine what the coefficients are on a taylor series expansion if the derivative is too hard to compute?

In a past lecture we talked about how you need to expand The Taylor series of a composed function based on what its input is. For, example: $e^u$ where ${\color{red} u} = \cos x=1 - \frac{1}{2!}x^2 ...
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1answer
188 views

Uniqueness of approximations like the Taylor polynomial

Given a function $f: \mathbb {R} ^n \to \mathbb {R} $, I am curious about the uniqueness of a $k$th-order approximation at $c \in \mathbb {R}^n $, i.e. a function $\phi(x)$ such that $$ \frac {f(c ...
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1k views

Taylor's Theorem for Multivariate Functions

Please look at this theorem in Wiki regarding Taylor's theorem generalized to multivariate functions: Multivariate version of Taylor's Theorem The version stated there is one that I'm not familiar ...
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83 views

What is a typical example of taylor series for $f(z)$ that converges iff $\operatorname{Re}(z)>0$

What is a typical example of taylor series for $f(z)$ that converges iff $\operatorname{Re}(z)>0$ where $\operatorname{Re}(z)>0$ is not a natural boundary ?
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76 views

$\frac{\sin x}{x^5} - \frac{1}{x^4} \underset{x\to 0}{\approx} \frac{-1}{6} \cdot \frac{1}{x^2}$, right?

I was reading an set of notes about Taylor series, and I came across a part I think is a typo. I want to make sure, because I want to understand this stuff correctly. Here is the relevant page of the ...
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2answers
552 views

Induction Proof for a series expansion of a function

I have done induction proofs of many different types, but trying to prove by induction that a derivative from the Taylor series expansion of a function has me stumped in terms of how to get the final ...
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1answer
27 views

Create function F() from Points

I would like to recreate a function only by knowing points on the graph. So I would have the points A(x/y) B(x/y) C(x/y) and would like to create its f() Is this possible? I heard this should be ...
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2answers
327 views

Short way? Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$

Write taylor's formula $f(x,y)=e^{xy}$ $a=(0,0)$ and $p=4$ Does there exist any short way? I have to calculate all partial dervatives. Is it?
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44 views

A question about Taylor polynomial

Let $f$ and $g$ be infinitely differentiable and the domian of $f$ equals to the domian of $g$ (says $D$). Then the following is true: If $f(x)=g(x)$ $\forall x$, then $f^{(n)}(a)=g^{(n)}(a)$ ...
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1answer
290 views

Taylor remainder of $f(x,y)=\sin x\cdot \cos y$

Given $f\colon \mathbb R^2\rightarrow\mathbb R,(x,y)\mapsto\sin x\cdot\cos y$ I want to show that there exists $M>0$ such that $$|f(x,y)-T_2(x,y)|\leq M(|x|+|y|)$$ for all $(x,y)\in\mathbb R^2$. ...
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513 views

Weighted uniform convergence of Taylor series of exponential function

Is the limit $$ e^{-x}\sum_{n=0}^N \frac{(-1)^n}{n!}x^n\to e^{-2x} \quad \text{as } \ N\to\infty \tag1 $$ uniform on $[0,+\infty)$? Numerically this appears to be true: see the difference ...
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Looking for examples where $f(z)=\operatorname{inv} \int_{0}^{z} g(z)\, dz$ with $f(z)$ entire and $g(z)$ not meromorphic.

I'm looking for examples where $f(z)=\operatorname{inv}\int_{0}^{z} g(z) \, dz$ with $f(z)$ entire and $g(z)$ not meromorphic. For clarity, by $\operatorname{inv}$, I mean the functional inverse. ...
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2answers
5k views

Taylor series expansion for $f(x)=\sqrt{x}$ for $a=1$

I seem to be stuck defining an alternating sequence of terms in this series because $f^{(0)}(x)=f(x)$ is positive, as well as $f'(x)$, but then every other term starting with $f''(x)$ is negative. How ...
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1answer
370 views

A problem related to mean value theorem and taylor's formula

I guess I need to use Taylor's formula and the mean value theorem. I have no idea except for them. Note: honestly, this is not homework. I am studying by myself. Suppose that ...
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1answer
74 views

Taylor expansion in $4D$

Let $f(x)=(x_2,-x_1,\sqrt 2 x_4 + x_1^3,-\sqrt 2x_3+x_3x_4^2)$ be a vector valued function from $\mathbb R^4\to\mathbb R^4$. Would anyone help me expand it up to and including the third term in its ...
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1answer
106 views

How do we know that $\sum_{k=0}^{\infty}\frac{x^k}{k!}=e^x$?

I've been taught that the definition of the exponential function is the following power series: $$\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ Here's my question: how do we know that this series is equal to ...
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1answer
145 views

Defining square root by series, and showing properties.

I'm triyng to define the root of a complex number near $1$ using the Taylor's series for $\sqrt{1+x}$, but I'm having some problems. Let $x\in\mathbb{C}$ such that $|x|<1$. Let ...
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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 ...
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3answers
363 views

Express $f(x) = x^2 \cos(2x) + \sin^2(x)$ as a power series

Express $f(x) = x^2 \cos(2x) + \sin^2(x)$ as a power series What I know: I know that $$x^2\cos(2x) = x^2 \cdot \sum_{n=0}^{\infty} {(-1)^n \cdot \frac{(2x)^{2n}}{(2n)!}} = \sum_{n=0}^{\infty} ...
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1answer
53 views

MacLaurin powerseries and interval of convergence

Given the function $f(x) = 5/(6*x^2-x-1)$, (a) Expand into MacLaurin powerseries the function $f$ up to order $3$. (b) Find the interval of convergence of it. (a) I will use the type of ...
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1answer
113 views

Asymptotic formula for complex gamma function at $+\infty+i \times y$

I am currently looking for the behaviour of the complex gamma function at real infinity: $\lim_{x \to \infty}\Gamma\left(x+i\times y\right)$ and more particularly for asymptotic formulas for the ...