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
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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
1k 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 ...
37
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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: ...
37
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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 $e$. ...
27
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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: ...
24
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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
564 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 ...
21
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1answer
359 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^{...
18
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2answers
183 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
307 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 ...
14
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3answers
591 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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6answers
7k views

Intuition explanation of taylor expansion?

Could you provide a geometric explanation of taylor expansion?
13
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7answers
11k 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}^\...
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
313 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
4k 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 ...
12
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2answers
538 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
4k 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 ...
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
190 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
613 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. ...
10
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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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4answers
309 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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1answer
640 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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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
164 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
750 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( e^{-\frac{x^2}{...
10
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1answer
461 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
905 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
399 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
644 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 ...
9
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3answers
81 views

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
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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
308 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 ...
9
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2answers
263 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
325 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) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
9
votes
1answer
711 views

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) = \...
9
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0answers
176 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)=...