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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3
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601 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 ...
1
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
103 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
146 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 ...
1
vote
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
202 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
469 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} ...
65
votes
3answers
1k 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
167 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
votes
0answers
45 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
490 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 ...
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1answer
29 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
494 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
169 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
745 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
183 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
66 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
976 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. ...
44
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
votes
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
58 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
68 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 ...
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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 $\sin{x}$, $e^x$?
1
vote
1answer
257 views

Determine whether a multi-variable limit exists

I need to determine whether the next limit exists: $$\lim_{(x,y)\to(0,0)}f(x,y)=\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$$ Looking at the numerator $(-1-\frac{x^2}2)$ it immediately ...
1
vote
1answer
80 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
114 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
88 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
72 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
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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
190 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
91 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 ...
1
vote
1answer
272 views

Laurent expansion of $\csc^2(\frac{\pi}{z})$ about $\frac{1}{3}$ for $|z-\frac{1}{3}| \lt \frac{1}{12}$

The question is: Find the Laurent expansion of $\csc^2(\frac{\pi}{z})$ about $\frac{1}{3}$ for $|z-\frac{1}{3}| \lt \frac{1}{12}$. In particular what is the coefficient of $(z-\frac{1}{3})^{-2}$. I ...
2
votes
1answer
77 views

Find Taylor series expansion and convergence radius for $\int_0^x\cos(\sqrt{t}\ )dt$

i must find the the Taylor series expansion (i've been asked not necessarily calculating it directly) and the convergence radios for this function : $$f(x) = \int_0^x \cos(\sqrt{t}\ ) \, dt$$ I am ...
1
vote
1answer
124 views

Exponential as power series

Is there a function that does not depend on $a$ such that $\sum_{x=1}^\infty \frac{a^x}{x!}f(x) = \mathrm e^{-a}$? Just to be clear, the summation starting from 1 is intentional, otherwise the ...
2
votes
1answer
68 views

Computing taylor series, getting all 0's

I started out by finding the first and second derivative. For $f'(x)$ I got $\;\;\dfrac{(12x^2-x^4)}{(4-x^2)^2}$ For $f''(x)$ I got $\;\;\dfrac{(4-x^2)(24x-4x^3)-(12x^2-x^4)(-4x) }{ (4-x^2)^3}$ ...
6
votes
3answers
94 views

Find $f^{(1001)}(0)$

I am to find the value in 0 of 1001th derivative of the function $$f(x) = \frac{1}{2+3x^2}$$ How should I approach this kind of problem? I tried something like : $$\frac{1}{2+3x^2} = ...
2
votes
2answers
178 views

What is the easiest/most efficient way to find the taylor series expansion of $e^{1-cos(x)}$ up to and including degrees of four?

So I have $$e^{1-cos(x)}$$ and want to find the taylor series expansions up to and including the fourth degree in the form of $$c_{0} \frac{x^0}{0!} + c_{1} \frac{x^1}{1!} + c_{2} \frac{x^2}{2!} + c_3 ...
0
votes
3answers
60 views

Calculate the sum $\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{n\times2^{2n+1}}$

I started with $arctg(x) = \sum\limits_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}$ Then I differentiated to get rid of the denominator. Then divide with $x$ to get $x^{2n-1}$. Then integrate to get ...
1
vote
1answer
47 views

Estimate the degree of a Taylor Polynomial using its Error Term

In my 2nd year studying Maths at Uni and revising for a Numerical Analysis final exam. We're given 1 past paper but no solutions, and I can't answer this question: Use the error term of a Taylor ...
0
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1answer
49 views

why start the taylor series of $\cos^{2} x$ at $k=1$ and not just $k=0$ as I do not understand the problem with $2^{-1}$

Im using $\cos^2 x=\frac{1}{2}(1+\cos(2x))$ and $\cos x = (-1)^k \frac{(2x)^{2k}}{(2k)!}$ to find the sum for the Taylor series of $\cos^2 x$. I thought I was getting it. When I find the answer ...
2
votes
3answers
71 views

Expand function into a Maclaurin's series

The function is given with: $f(x)=\dfrac{x^{2012}}{(1-x^3)^2}$ I have no idea what to do here and honestly, I don't really understand what a Maclaurin's series is. I know the definition but I don't ...
4
votes
2answers
90 views

Why can't you find all antiderivatives by integrating a power series?

if $f(x) = \sum\limits^{\infty}_{n=0}\frac{f^{(n)}(0)}{n!}x^n$ why can't you do the following to find a general solution $F(x) \equiv \int f(x)dx$ $F(x) = \int ...
4
votes
1answer
137 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 ...
5
votes
1answer
233 views

Maclaurin series of $f(x)=\sinh(1/x)$?

As we know the formula of Maclaurin series for $f(x) = \sinh(x)$ is $f(x)=x+x^3/3! + x^5/5!+\ldots$ Could anyone tell me what is the Maclaurin series of $f(x)=\sinh(1/x)$?
3
votes
3answers
607 views

Trigonometric identities using $\sin x$ and $\cos x$ definition as infinite series

Can someone show the way to proof that $$\cos(x+y) = \cos x\cdot\cos y - \sin x\cdot\sin y$$ and $$\cos^2x+\sin^2 x = 1$$ using the definition of $\sin x$ and $\cos x$ with infinite series. thanks...
1
vote
1answer
685 views

Maclaurin series for sin(x) representation

The Maclaurin series for $\sin(x)$ is: $$ \sin(x) = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} ... $$ Which according to wikipedia is: $$ \displaystyle \sum_{n=0}^{\infty} ...
3
votes
1answer
113 views

When computing the Taylor series of $(\cos x)^2$ how does the slide jump to concluding it is $1-(\sin x)^2$?

In the following slide it shows how the taylor series of $(\cos x)^2$ is computed: On the first line they simply take the taylor series of cosx and write it out twice, which makes sense. However, ...