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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2
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2answers
450 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 ...
1
vote
1answer
26 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 ...
0
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2answers
44 views

$R_{n}\left ( x \right )$ for $ f\left ( x \right) = \cos\left ( 2\cdot x \right ) $?

How can I define $R_{n}\left ( x \right )$ for $$ f\left ( x \right) = \cos\left ( 2\cdot x \right ) $$ I found taylor expansion for $cos2x$.What should I do after that? My problem is I dont know if ...
1
vote
2answers
296 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?
0
votes
2answers
43 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 $f^{(n)}(a)=g^{(n)}(a)$ ...
2
votes
1answer
287 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 $(x,y)\in\mathbb R^2$. ...
8
votes
3answers
487 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 ...
0
votes
1answer
72 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 ...
1
vote
0answers
56 views

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. ...
2
votes
2answers
154 views

Taylor series expansion and approximation

I found this amazing question in the last calculus exam, but I don't know how to answer. Let $T(x) = \ln(1+a) + \frac{1}{1+a}(x-a) - \frac{1}{2(1+a)^2}(x-a)^2 +...+ ...
1
vote
1answer
62 views

How large must $n$ be if $|\sin(x) - T_n(x)| < 0.01$ for all $x\in[-\pi,\pi]$?

Suppose that $f(x) = \sin(x)$, and that $T_n(x)$ is the $n$th Taylor's polynomial of $f$ centered at $0$. How large must $n$ be if $|f(x) - T_n(x)| < 0.01$ for all $x$ on the interval ...
1
vote
1answer
332 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 ...
0
votes
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 ...
1
vote
1answer
132 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 ...
2
votes
3answers
327 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} ...
1
vote
1answer
108 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 ...
1
vote
1answer
44 views

Rearranging power series expansion to get parameter on denominator

How can we rearrange $$T=\dfrac{k V+g}{gk}\bigg(kT-\dfrac{1}{2}k^{2}T^{2}+\dfrac{1}{6}k^{3}T^{3}\bigg),$$ to get $$T=\dfrac{2V/g}{1+k V/g}+\dfrac{1}{3}k T^{2}$$ ?
3
votes
3answers
662 views

Infinite (Taylor) Series

What is the general term for a Taylor series of $\sin(x)$ centered at $\pi/4$? It should be $(-1)^{[??]} \times \sqrt{2}/2 \times \frac{(x-\pi/4)^n}{n!}$ What power is $(-1)$ supposed to be raised ...
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vote
0answers
115 views

Can we compute Fourier series of any function this way?

There is a technique to compute Fourier series much quickly, but I doubt how general this technique can be. Let's look at a simple example to see how the technique goes. Compute Fourier series of the ...
3
votes
1answer
108 views

Differentiate a hypergeometric function expression

I have the following function $$f_\epsilon (p)=\frac{1}{2}(1-p)^\epsilon 2^\epsilon {_2}F_1(1-\epsilon,\epsilon;1+\epsilon;\frac{1-p}{2}),\qquad p\in(-1,1).$$ Here $F$ is the hypergeometric ...
2
votes
1answer
188 views

Numerical Analysis best estimate on polynomial order

I need to determine the best integer value of $k$ for the equation: \begin{equation} \arctan(x) = x + O(x^k) \text{ as $x\to 0$} \end{equation} Taylor's Theorem with Lagrange Remainder would ...
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3answers
69 views

Numerical Analysis and Big O

How can I show that $e^x -1$ is not $O(x^2)$ as $x\to0$ I'm not sure where to start. We can use Taylor's Theorem with remainder: \begin{equation} e^x = \sum\limits_{k=0}^n\dfrac{x^n}{n!} ...
3
votes
0answers
232 views

Taking a Fourier transform of Taylor series

My (naive) question is whether it is possible to take the Fourier transform of a Taylor series? Could one use multiplication with $\delta$ to get the function sampled at the point of expansion and ...
4
votes
4answers
515 views

Find nth derivative of $\frac{x^{n}}{(1-x)^{2}}$, please?

I need to find the nth derivative of $\frac{x^{n}}{(1-x)^{2}}$ for $0<x<1$ So far, I tried the same method used for $\frac{x^{n}}{1-x}$ and here's what I got: \begin{equation} ...
67
votes
3answers
2k views

Is the derivative the natural logarithm of the left-shift?

(Disclaimer: I'm a high school student, and my knowledge of mathematics extends only to some elementary high school calculus. I don't know if what I'm about to do is valid mathematics.) I noticed ...
2
votes
0answers
198 views

Proving Lagrange's Remainder of the Taylor Series

My text, as many others, asserts that the proof of Lagrange's remainder is similar to that of the Mean-Value Theorem. To prove the Mean-Vale Theorem, suppose that f is differentiable over $(a, b)$ ...
0
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0answers
46 views

How to choose a numeric approach for derivates

I would like to find the derivate from some combined logistic and exponential functions that all describe the same data numerically. $$f'(t)=\frac{f(t+h)-f(t)}{h}$$ seems not the best choise for ...
10
votes
4answers
522 views

Show that $e^x \geq (3/2) x^2$ for all non-negative $x$

I am attempting to solve a two-part problem, posed in Buck's Advanced Calculus on page 153. It asks "Show that $e^x \geq \frac{3}{2}x^2$ $\forall x\geq 0$. Can $3/2$ be replaced by a larger ...
0
votes
1answer
30 views

Prove that the polynomial divided by a fraction of the power of n is equal to the sum of fractions of any constans and successive powers of

Let n≥1 and n is integer. P(x) - polynomial and $deg P(x)<n$. Prove if $ a \in \Bbb R/{0} $ then: $ \frac{P(x)}{(ax+b)^n} = \frac{c_1}{ax+b} + \frac{c_2}{(ax+b)^2}+...+\frac{c_n}{(ax+b)^n}$ for ...
11
votes
2answers
508 views

Taylor series (or equivalent at $\epsilon\to0$) of the integral over $x$ of a function of $x$ and $\epsilon$

I have a function $f$ of two arguments, defined as $$ f(x,\epsilon)=\epsilon\left( e^{-\frac{(x-\epsilon)^2}{2}} - e^{-\frac{x^2}{2}}\right) + \frac{1-\epsilon}{\sqrt{1+\epsilon}}\left( ...
2
votes
2answers
83 views

failed application of magicry in Taylor expansion of $1/x^2$ near $x=2$

It's straightforward to find the Taylor expansion for $\frac{1}{x^2}$ near $x=2$ using the the Taylor series definition. This is turns out to be $\frac{1}{4} - \frac{1}{4} (x-2) + \frac{3}{16}(x-2)^2 ...
0
votes
1answer
212 views

Confused about a limit proof and Big O.

I gave an incorrect proof here : How can evaluate $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$ I am confused as when considering the mistakes in my proof it seems the limit cannot be ...
0
votes
1answer
983 views

estimating the error of $\sin(x) = x$ with Taylor's Theorem

I want to calculate the numerical error in approximating $\sin(x)=x$ with Taylor's Theorem. Furthermore, what values of $x$ will this approximation be correct to within $7$ decimal places? Here is ...
0
votes
3answers
211 views

Taylor series of $ f = e^{x^2 + y^2}$ near $(0,0)$

I have to compute the second order Taylor series of the function $ f = e^{x^2 + y^2}$ near $(0,0)$. The Jacobian is: $$ Df(x,y) = (2\ x\ e^{x^2 + y^2}, 2\ y\ e^{x^2 + y^2}) $$ and the Hessian: $$ ...
1
vote
1answer
67 views

Estimate the scale of the power series with Poisson pdf-like terms

Sorry to bother you, but I guess that this question is not appropriate for MO, so I repost it here hoping that someone could give me a clue. I would like to have an estimate for the series $$P(t) = ...
8
votes
1answer
1k views

Taylor Series and Fourier Series

Taylor series expansion of function, $f$, is a vector in the vector space with basis: $\{(x-a)^0, (x-a)^1, (x-a)^3, \ldots, (x-a)^n, \ldots\}$. This vector space has a countably infinite dimension. ...
46
votes
4answers
7k views

Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. My question: What is the connection between these two? Is there a way to get from one to the other (and ...
0
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0answers
38 views

How to Taylor expand $\ln{1-\exp{-i_t}}$ around i?

my question here is how to Taylor expand around $i$ $\ln{(1-\exp{(-i_t)})}$ to the first order? $i_t$ is a time series variable, $i$ is its steady state. Could anyone show me how to expand it ...
2
votes
0answers
59 views

Taylor polynomial of $\frac{1}{1-x-y}$

I need to calculate the 2nd order Taylor polynomial at the origin of $$f(x,y) = {1 \over{1-x-y}}$$ I have looked at two ways, and not sure which is simpler. We can split it by partial derivatives ...
0
votes
1answer
69 views

Differentiation term by term of Taylor series

Suppose I have A Taylor Series of a function around $z_{0}$ in the complex plane which convergence in a ball of radius $r>0$. Can I differentiate term by term the Taylor series and get the ...
0
votes
1answer
27 views

Taylor of $f:\Bbb R^3\to \Bbb R$

My notes say the following: You have a function $D(x, y, \sigma)$ mapping to a scaler. Take the taylor expension (which I can only do for functions from $\mathbb{R} \to \mathbb{R}$) up to the ...
1
vote
3answers
74 views

Find all the numbers $x$ such that $\sum_{n=0}^{\infty} \frac{x^n}{(2n)!}=0$

Find all the numbers $x$ such that $$\sum_{n=0}^{\infty} \frac{x^n}{(2n)!}=0$$ Is it by some tricks on Taylor series on $\sin{x}$, $e^x$?
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1answer
86 views

trouble expanding taylor series about a point other than zero using geometric series

I'm trying to understand how to use a Taylor series expansion to correctly expand a population growth function about a point other than zero using the geometric series. For expansion about $t=0$, I ...
2
votes
1answer
123 views

Taylor Series Remainder

Use Taylor's Theorem to estimate the error in approximating $\sinh 2x$ by $2x + 4/3x^3$ on the interval $[-0.5,0.5]$. For this question, I use the Taylor's remainder formular, $$ R_n(x)= ...
2
votes
1answer
97 views

Using Maclaurin series with solving a multi-variable limits

I need to determine wheter there's a limit where $(x,y)=(0,0)$ of the next function: $$\lim_{(x,y)\to(0,0)}\frac{e^{x(y+1)}-x-1}{\sqrt{x^2+y^2}}$$ In order to simplify the expression can I use ...
0
votes
1answer
81 views

Expand log function with two terms

HOw can I expand ln(1+2/(A-1))? I think I need to use taylor series but the 1 is messing me up. Should I just ignore the 1?
0
votes
3answers
79 views

Taylor's Remainder

what is the maximum error when approximating $e^{x}$ by $1+x+\frac{x^{2}}{2}$ for $|x|<1$? Answer for this is $\frac{e}{6}$. Can anyone teach me the working for this question, please?
1
vote
2answers
39 views

How to evaluate binomial coefficients when $k=0$ and $1\geq|n|\geq0$

So I am doing some taylor series which boil down to the geometric series, for which I then need to evaluate various binomial coefficients. I have always used $n\choose k$ $=\frac{n!}{(n-k)!k!}$ to do ...
4
votes
1answer
195 views

Extending partial sums of the Taylor series of $e^x$ to a smooth function on $\mathbb{R}^2$?

Is there a smooth function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,n)$, where $n\in\mathbb{N}$, is the truncated Taylor series of $e^x$, namely $1+ x + \frac{x^2}{2} + \dotsb + \frac{x^n}{n!}$, ...
0
votes
1answer
94 views

Maclaurin vs Taylor and their geometrical difference

In this topic i learned how to approximate a function with a high degree polynomial and how to derive the Maclaurin series: $$ f (x) = P_n(x) = f(0)+{f'(0)\over 1!}x+{f''(0)\over ...