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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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 ...
73
votes
6answers
13k 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 ...
44
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2answers
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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 ...
39
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13answers
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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: ...
38
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3answers
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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\...
35
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3answers
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$e$ 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 $e$. ...
29
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9answers
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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: ...
25
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2answers
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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 ...
22
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1answer
369 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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1answer
565 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 ...
18
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2answers
185 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 ...
18
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1answer
315 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 $...
17
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1answer
341 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 ...
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6answers
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Intuition explanation of taylor expansion?

Could you provide a geometric explanation of taylor expansion?
14
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7answers
12k 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}^\...
14
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3answers
606 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, ...
13
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4answers
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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 $I$? ...
13
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1answer
315 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 ...
12
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3answers
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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 ...
12
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2answers
541 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{...
12
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1answer
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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 ...
12
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3answers
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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 ...
12
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4answers
194 views

Nested radical $\sqrt{x+\sqrt{x^2+\cdots\sqrt{x^n+\cdots}}}$

I am studying the $f(x) = \sqrt{x+\sqrt{x^2+\cdots\sqrt{x^n+\cdots}}}$ for $x \in (0,\infty)$ and I am trying to get closed form formula for this, or at least some useful series/expansion. Any ideas ...
11
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5answers
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Why do un-integrable functions exist?

By un-integrable I mean functions whose antiderivative can not be expressed in terms of elementary functions. I recently learnt that any differentiable function can be expanded using the Taylor ...
11
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3answers
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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\...
11
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2answers
621 views

Application of Taylor's Theorem in Number Theory

I'm working through Alan Baker's book A Concise Introduction to the Theory of Numbers, and there's an assertion in there that confuses me. Here's the quote: It is easily seen that no polynomial $f(...
11
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6answers
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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 ...
11
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1answer
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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. ...
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7answers
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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 $\sqrt{x}$...
10
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4answers
573 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 constant?"...
10
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4answers
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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 ...
10
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5answers
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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\...
10
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1answer
645 views

Basic Taylor expansion question

I seem to have a misunderstanding of how to work with a Taylor series. Suppose I want to write $f(x)=x e^x$'s Taylor expansion of $n$ degree around $0$. I see two ways: 1) Find the $n$th ...
10
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4answers
310 views

Which expansion of $e$ is more accurate?

We have two forms of $e^x$ $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....$$ and $$e^x=\frac{1}{\displaystyle 1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+....}$$ The second form comes from $e^x=1/e^{-x}...
10
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3answers
483 views

Faster way to find Taylor series

I'm trying to figure out if there is a better way to teach the following Taylor series problem. I can do the problem myself, but my solution doesn't seem very nice! Let's say I want to find the ...
10
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3answers
269 views

Quick way to expand $\cos^{-1}(\cos^2 x)$ up to $O(x^2)$

For a part of a question, I need to expand $\cos^{-1}(\cos^2 x)$ up to $O(x^2)$ about $x=0$. It took me quite a while to get an incorrect answer. What are some quick and efficient offline (i.e, no ...
10
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4answers
168 views

Bernoulli Number analog using Cosine

I know that Bernoulli Numbers can be found with the generating function $$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$ I was wondering if any work has been done using a similar equation $$\...
10
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2answers
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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( e^{-\frac{x^2}{...
10
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2answers
323 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 ...
10
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1answer
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The error term in Taylor series and convolution.

I've been wondering a lot why is the remainder of the Taylor expansion of a function, $R_n(x)$, expressed (in one of the many forms) as something very similar to aconvolution. Precisely: $$R_n(x) = \...
10
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1answer
464 views

Old versus New enunciation of Taylor's Theorem.

I am studying from Spivak' Calculus, and he states Taylor's Theorem as follows: THEOREM Let $f',\cdots,f^{(n+1)}$ be defined on $[a,x]$ and let $R_{n,a}(x)$ be defined by $$R_{n,a}(x)=f(...
9
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6answers
908 views

Why the existence of Taylor series doesn't imply it coverges to the original function

Please note that I've read this question and it did not address mine. I've been presented with the following argument regarding Taylor series: We have a function $f(x)$, now assume that there ...
9
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3answers
401 views

difference of square roots approximation

In two of my physics courses in the past week, I've come across an approximation for the difference of two square roots for large radicands: $\sqrt{x+a}-\sqrt{x+b}\approx\frac{a-b}{2\sqrt x}$ for ...
9
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4answers
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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 ...
9
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2answers
651 views

Euler's identity in matrix form

I assume everyone is familiar with the famous mathematical identity due to L. Euler: $$ e^{i \, \pi} + 1 = 0,$$ where $i^2 = -1$ and $e$ is the base of natural logarithms. I was wondering if this ...
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3answers
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Prove that $\cosh^{-1}(1+x)=\sqrt{2x}(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+…)$

How can we prove the series expansion of $$\cosh^{-1}(1+x)=\sqrt{2x}\left(1-\frac{1}{12}x+\frac{3}{160}x^2-\frac{5}{896}x^3+...\right)$$ I know the formula for $\cosh^{-1}(x)=\ln(x+\sqrt{x^2-1})$...
9
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2answers
3k views

Practical Use of Series Expansion at $x=\infty$

Asking WolframAlpha on certain functions, it happens that you get a series expansion at $\infty$. Thinking of the expansion as an approximation of the function in the vincinity of a point $a$, like in ...
9
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2answers
265 views

Equivalence of $\pi$ is the first positive zero of the taylor series for $\sin(x)$ and $\pi/4 = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$

For $x\in\mathbb{R}$, define $\sin (x) = x - x^3/3!+x^5/5!-\cdots$ and $\pi = 4(1-\frac{1}{3}+\frac{1}{5} -\frac{1}{7}+\cdots)$. Then show that $\sin(\pi/2) = 1$ In the prologue of Real and Complex ...
9
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3answers
149 views

Prove that $\lim_{x\to\infty} f'(x) = 0$ [duplicate]

Let $f(x)$ be twice differentiable on $(0,\infty)$ and let $\lim_{x\to \infty} f(x) = L<\infty$ and $|f''(x)| \le M$ for some $M>0$. Prove that $\lim_{x \to \infty} f'(x) = 0$. I've tried to ...
9
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1answer
326 views

Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given converging series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...