Questions tagged [taylor-expansion]

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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119
votes
6answers
24k views

Connection between Fourier transform and Taylor series

Both Fourier transform and Taylor series are means to represent functions in a different form. What is the connection between these two? Is there a way to get from one to the other (and back again)? ...
98
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3answers
3k 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 ...
63
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14answers
123k views

What are the practical applications of the Taylor Series?

I started learning about the Taylor Series in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series. Question: ...
57
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11answers
42k views

Simplest proof of Taylor's theorem

I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem. By which I mean this: there are plenty of proofs that introduce some arbitrary construct: ...
48
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11answers
5k views

Why is the notion of analytic function so important?

I think I have some understanding of what an analytic function is — it is a function that can be approximated by a Taylor power series. But why is the notion of "analytic function" so important? I ...
48
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2answers
2k views

Is there a function with the property $f(n)=f^{(n)}(0)$?

Is there a not identically zero, real-analytic function $f:\mathbb{R}\rightarrow\mathbb{R}$, which satisfies $$f(n)=f^{(n)}(0),\quad n\in\mathbb{N} \text{ or } \mathbb N^+?$$ What I got so far: Set ...
40
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3answers
2k views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know $$\underset{j=a}{\overset{b}{\LARGE\...
37
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3answers
2k views

$e$ to 50 billion decimal places

Sorry if this is a really naive question, but in my reading of a lot of textbooks and articles, there is a lot of mention of how many decimals we know of a certain number today, such as $\pi$ or $e$. ...
33
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0answers
694 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider $f(x)=...
30
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8answers
59k views

How are the Taylor Series derived?

I know the Taylor Series are infinite sums that represent some functions like $\sin(x)$. But it has always made me wonder how they were derived? How is something like $$\sin(x)=\sum\limits_{n=0}^\...
28
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2answers
1k views

How local is the information of a derivative?

I have read it a thousand times: "you only need local information to compute derivatives." To be more precise: when you take a derivative, in say point $a$, what you are essentially doing is taking a ...
25
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3answers
16k views

Why doesn't a Taylor series converge always?

The Taylor expansion itself can be derived from mean value theorems which themselves are valid over the entire domain of the function. Then why doesn't the Taylor series converge over the entire ...
25
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1answer
531 views

Function $f(x)$, such that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$

Consider a function $f(x)$. Define Taylor series $\sum_{n=0}^{\infty} f(n) x^n$. Is there such a function, other than constant $0$, that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$? The Taylor series of $...
23
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7answers
64k views

Taylor series for $\sqrt{x}$?

I'm trying to figure Taylor series for $\sqrt{x}$. Unfortunately all web pages and books show examples for $\sqrt{x+1}$. Is there any particular reason no one shows Taylor series for exactly $\sqrt{x}$...
23
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1answer
484 views

Are there always singularities at the edge of a disk of convergence?

Take a function that is analytic at 0 and consider its Maclaurin Series. Here are some examples I'll refer to: $$\frac{1}{1-x} =\sum_{n=0}^\infty x^n$$ $$\frac{1}{1+x^2} =\sum_{n=0}^\infty(-1)^nx^{...
22
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6answers
17k views

An intuitive explanation of the Taylor expansion

Could you provide a geometric explanation of the Taylor series expansion?
22
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2answers
285k views

taylor series of $\ln(1+x)$?

Compute the taylor series of $\ln(1+x)$ I've first computed derivatives (up to the 4th) of ln(1+x) $f^{'}(x)$ = $\frac{1}{1+x}$ $f^{''}(x) = \frac{-1}{(1+x)^2}$ $f^{'''}(x) = \frac{2}{(1+x)^3}$ $f^...
22
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3answers
18k views

What is the difference between the Taylor and Maclaurin series?

What is the difference between the Taylor and the Maclaurin series? Is the series representing sine the same both ways? Can someone describe an example for both?
22
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3answers
327 views

Do the Taylor series of $\sin x$ and $\cos x$ depend on the identity $\sin^2 x + \cos^2 x =1$?

I had this crazy idea trying to prove the Pythagorean trigonometric identity;$$\sin^2x+\cos^2x=1$$by squaring the infinite Taylor series of $\sin x$ and $\cos x$. But it came out quite beautiful, ...
21
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3answers
1k views

Understanding the Taylor expansion of a function

Suppose $$f(x) = \frac{1}{1+x^2}$$ We know this function is defined everywhere and is continuous everywhere and so on... Using the geometric series, we can write $$ \frac{1}{1+x^2} = \sum (-x^2)^...
21
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3answers
2k views

Settle a classroom argument - do there exist any functions that satisfy this property involving Taylor polynomials?

I'm going to apologize in advance; I might at some points say Taylor series instead of Maclaurin series. OK, so backstory: My calculus class recently went over Taylor series and Taylor polynomials. ...
21
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1answer
613 views

Is the Maclaurin series expansion of $\sin x$ related to the inclusion-exclusion principle?

When I see the alternating signs in the infinite series expansion of $\sin x$, I'm reminded of the inclusion-exclusion principle. Could there be any way to visualize it in such a way? Also, is there ...
20
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6answers
5k views

When do Taylor series provide a perfect approximation?

To my understanding, the Taylor series is a type of power series that provides an approximation of a function at some particular point $x=a$. But under what circumstances is this approximation perfect,...
20
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2answers
7k views

Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

What is the (simple) closed form for $\large \displaystyle S(m) = \sum_{n=1}^\infty \dfrac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$? Notation: $ \dbinom{2n}n $ denotes the central binomial ...
19
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3answers
5k views

Is Fourier series an “inverse” of Taylor series?

I've understood Taylor series as being the representation of a "transcendental" function, using power functions with coefficents represented by appropriate derivatives. (Or maybe it is the MacLauren ...
19
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5answers
1k views

Third degree Taylor series of $f(x) = e^x \cos{x} $

Suppose you have the function: $$f(x) = e^x \cos{x} $$ and you need to find the 3rd degree Taylor Series representation. The way I have been taught to do this is to express each separate function as ...
19
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1answer
10k views

On the vector spaces of 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. ...
18
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4answers
30k views

Are Taylor series and power series the same “thing”?

I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general ...
18
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1answer
2k views

Taylor series leads to two different functions - why?

Suppose, I want to find a function such that its Taylor series expansion is $$f(x) = \sum_{n=0}^{\infty}\frac{x^{n+1}}{(n+1)a^n}$$ I could start with $$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n$$ ...
18
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2answers
266 views

Is it possible to detect periodicity of an analytic function from its Taylor series coefficients at a point?

Given the Taylor series $\sum a_k (x - x_0)^k$ of an analytic function, it is possible to determine whether the function is periodic more-or-less directly from the coefficients $a_0, a_1, \ldots$ of ...
17
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3answers
1k views

Under what conditions is integrating over a series expansion valid for an improper integral?

On stackoverflow, a question was asked about getting Mathematica to evaluate the integral, $$\int^\infty_0 \frac{e^{-x}}{\sin x} \, \mathrm{d}x$$ which we know is divergent. In one of the answers, ...
17
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1answer
406 views

$x^3-3x-3=0$, prove that $10^x<127$

$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$ I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
17
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1answer
9k 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 ...
15
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6answers
16k views

How do Taylor polynomials work to approximate functions?

I (sort of) understand what Taylor series do, they approximate a function that is infinitely differentiable. Well, first of all, what does infinitely differentiable mean? Does it mean that the ...
15
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1answer
2k views

Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

If you map the nth roots of unity $z$ with the function $-W(-z/e)$ you get decent starting points for some root finding algorithm to the roots of the scaled truncated taylor series of $\exp$. Here W ...
15
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3answers
3k views

Taylor expansion of $(1+x)^α$ to binomial series – why does the remainder term converge?

For $α ∈ ℝ$ the function $g_α \colon B_1(0) → ℝ, x ↦ (1+x)^α$ is $C^∞$ and $g_α^{(n)}(x) = n! \tbinom{α}{n}(1+x)^{α-n}$, where $\tbinom{α}{n} = \frac{α(α-1)\cdots(α-n+1)}{n!}$ is the generalized ...
15
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0answers
290 views

Geometric representation of Euler-Maclaurin Summation Formula

When reading Tom Apostol's expository article (or the free link), I was expecting more diagrams to come that follow the figure below (or this from the Wolfram project). It was a disappointment not ...
14
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4answers
7k views

On what interval does a Taylor series approximate (or equal?) its function?

Suppose I have a function $f$ that is infinitely differentiable on some interval $I$. When I construct a Taylor series $P$ for it, using some point $a$ in $I$, does $f(x) = P(x)$ for all $x$ in $I$? ...
14
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4answers
4k views

How many smooth functions are non-analytic?

We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather ...
14
votes
2answers
961 views

How to compute the values of this function ? ( Fabius function )

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...
14
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2answers
943 views

What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
14
votes
5answers
4k views

Substituting for Taylor series

So my question is simple: Why is substitution valid? I mean it seems counter-intuitive to me mainly because of the chain rule. For example: The Taylor series of $e^{x^2}$ is simply done by ...
13
votes
7answers
16k views

Multivariate Taylor Expansion

I am in confidence with Taylor expansion of function $f\colon R \to R$, but I when my professor started to use higher order derivatives and multivariate Taylor expansion of $f\colon R^n \to R$ and $f\...
13
votes
3answers
2k views

Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?

Using the Taylor expansion $$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain $$f(x+a) = e^{a\...
13
votes
2answers
1k views

Elementary Proof of Ramanujan Master Theorem

I was searching for an elementary proof of the Ramanujan Master Theorem and I found a page from Ramanujan's Notebook on wikipedia which contained the proof. I think that it has some gaps, so can ...
13
votes
3answers
5k views

How to check the real analyticity of a function?

I recently learnt Taylor series in my class. I would like to know how is to possible to distinguish whether a function is real-analytic or not. First thing to check is if it is smooth. But how can I ...
13
votes
3answers
658 views

A property of roots of the truncated series for $\sin(x)$

Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$ In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$. $\begin{...
13
votes
1answer
5k views

Clever derivation of $\arcsin(x)$ Taylor series

I was working the other day in the Math Help Centre, trying to help some first years with a calculus problem. The problem involved investigating the Taylor series of $\arcsin(x)$. Once the students ...
13
votes
3answers
602 views

If $f^2$ and $f^3$ are $C^{\infty}(\mathbb R)$ then $f$ is $C^{\infty}(\mathbb R)$

Since it is an exercise from an oral exam, I have added some indications I had. $f : \mathbb R \to \mathbb R$, such that $f^2$ and $f^3$ are $C^{\infty}$, show that $f$ is $C^{\infty}$. The two ...
13
votes
1answer
404 views

Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the ...