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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68
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6answers
12k 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 ...
3
votes
4answers
1k views

Formula for calculating $\sum_{n=0}^{m}nr^n$

I want to know the general formula for $\sum_{n=0}^{m}nr^n$ for some constant r and how it is derived. For example, when r = 2, the formula is given by: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
37
votes
13answers
49k 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: ...
2
votes
3answers
4k views

How to expand $\tan x$ in Taylor order to $o(x^6)$

I try to expand $\tan x$ in Taylor order to $o(x^6)$, but searching of all 6 derivative in zero (ex. $\tan'(0), \tan''(0)$ and e.t.c.) is very difficult and slow method. Is there another way to ...
14
votes
6answers
7k views

Intuition explanation of taylor expansion?

Could you provide a geometric explanation of taylor expansion?
11
votes
3answers
1k 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) = ...
44
votes
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 ...
9
votes
4answers
157 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 ...
8
votes
1answer
239 views

Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?

Im looking for a real-analytic function $f(z)$ such that for any $z$ $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number and where $z$ is close to , or onto the real line such that $z$ is in the ...
6
votes
4answers
368 views

Why do the endpoints of the Maclaurin series for arcsin converge?

The series $$\sum_{n=0}^\infty {{-\frac {1} 2} \choose n} \frac{(-1)^n}{2n+1}$$ is an endpoint for the Maclaurin series for arcsin(x). (The other endpoint is just the negative of this one.) I played ...
1
vote
1answer
52 views

Find the residue at $z=-2$ for $g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$

Find the residue at $z=-2$ for $$g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ I know that: $$\psi(z+1) = -\gamma - \sum_{k=1}^{\infty} (-1)^k\zeta(k+1)z^k$$ Let $z \to -1 - z$ to get: $$\psi(-z) = ...
3
votes
2answers
26k views

taylor series of ln(1+x)?

Compute the taylor series of $ln(1+x)$ I've first computed derivatives (upto the 4th) of ln(1+x) $f^{'}(x)$ = $\frac{1}{1+x}$ $f^{''}(x) = \frac{-1}{(1+x)^2}$ $f^{'''}(x) = \frac{2}{(1+x)^3}$ ...
1
vote
1answer
166 views

Integral remainder converges to 0

I want to show that $\displaystyle \log(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$ for $-1<x\leq1$ i want to show it with the integral remainder of the taylor series that gave me: ...
21
votes
1answer
348 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} ...
9
votes
4answers
2k 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 ...
9
votes
1answer
699 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) = ...
4
votes
3answers
1k views

Taylor series for different points… how do they look?

I can't understand what it means to do the Taylor series at the point $a$. The best way would be showing me how it looks for different $a$ on a graph. Do I find those graphs on the Internet?
1
vote
1answer
191 views

An issue with approximations of a recurrence sequence

By trying to give an approximation to a given recurrence sequence I encountered a problem. To be more precise I have a method but it fails if the right condition is not met and I wonder how I should ...
7
votes
2answers
307 views

How to prove $\int_0^1 \frac{1+x^{30}}{1+x^{60}} dx = 1 + \frac{c}{31}$, where $0 < c < 1$

This is an exercise from Apostol (p.285) that I'm having trouble with (in fact, I'm having trouble with the whole section): Prove that $\displaystyle{\int_0^1 \frac{1+x^{30}}{1+x^{60}} = 1 + ...
3
votes
1answer
752 views

Taylor series for logarithm converges towards logarithm

Is there a way to show that the Taylor series around 0 of $f(x) = \ln(1-x)$ converges towards $f$ on the interval $(-1,1)$, just by considering the remainder from the Taylor polynomial? I'm having a ...
3
votes
1answer
2k views

Generalized binomial theorem

Prove that: $$(1+x)^{\alpha}=\sum_{n=0}^{+\infty}{\alpha \choose n} x^n$$ for $x\in[0;1), \alpha \in\mathbb{R}$ based on Taylor's theorem with Lagrange remainder. I don't feel such proofs. ...
0
votes
1answer
166 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
8
votes
3answers
632 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 ...
1
vote
1answer
141 views

Complex Taylor and Laurent expansions

Let $f(z):=\dfrac{1}{2-z-z^2}, z\in\mathbb{C}\setminus\left\{ {1, -2}\right\}$. i) Express $f$ in the form $\dfrac{A}{1-z}+\dfrac{B}{2+z}$. [Answer to this is $\dfrac{1/3}{1-z}+\dfrac{1/3}{2+z}$]. ...
0
votes
1answer
3k 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 ...
32
votes
1answer
1k 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 ...
8
votes
1answer
118 views

Taylor expansion for $\arcsin^2{x}$

I stumbled upon this particular expansion that was included in this post. $$ \displaystyle \arcsin^{2}(x) = \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n^{2} \binom{2n}{n}} (2x)^{2n}$$ This caught ...
5
votes
1answer
82 views

Evaluate the improper integral $ \int_0^1 \frac{\ln(1+x)}{x}\,dx $

I am trying to evaluate $$ \int_0^1 \frac{\ln(1+x)}{x}\,dx $$ I started by using the Taylor series for $\ln (1+x)$ $$\begin{align*} \int_0^1 \frac{\ln(1+x)}{x}\,dx &= ...
12
votes
3answers
1k 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 ...
6
votes
1answer
2k views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
5
votes
3answers
715 views

Why does this infinite series equal one?

Why does $$\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$$ Is there an intuitive method by which to derive this equality?
3
votes
2answers
293 views

Taylor Series of $\tan x$

I found a nice general formula for the Taylor series of $\tan x$: $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$ where $B_n$ are the Bernoulli ...
8
votes
7answers
2k 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 ...
5
votes
5answers
352 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
2
votes
5answers
3k views

What's the Maclaurin Series of $f(x)=\frac{1}{(1-x)^2}$?

This function seemed to be pretty much straight forward, but my solution is incorrect. I have two questions: 1. Where did I make a mistake? 2. I learned that there are shortcuts for finding a series ...
8
votes
2answers
877 views

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

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...
4
votes
0answers
317 views

Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
1
vote
1answer
151 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
-2
votes
1answer
8k views

How does one approximate $\cos(58^\circ)$ to four decimal places accuracy using Taylor's theorem?

When one needs to compute say $\cos (58^\circ)$ with an error of at most $10^{-4}$, how does one go about it? What is an appropriate centre of the Taylor expansion, and how does one determine the ...
8
votes
2answers
268 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. ...
5
votes
2answers
78 views

Examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$

How to examine convergence of $\sum_{n=1}^{\infty}(\sqrt[n]{a} - \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2})$ for $a, b, c> 0$ using Taylor's theorem?
4
votes
0answers
266 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} ...
4
votes
2answers
163 views

Intuition behind Taylor/Maclaurin Series

** This is a different question than Intuition explanation of taylor expansion? ** I understand some of the intuition behind a Taylor/Maclaurin expansion. More specifically, I understand that adding ...
4
votes
2answers
201 views

How do I obtain the Laurent series for $f(z)=\frac 1{\cos(z^4)-1}$ about $0$?

I know that $$\cos(z^4)-1=-\frac{z^8}{2!}+\frac{z^{16}}{4!}+...$$ but how do I take the reciprocal of this series (please do not use little-o notation)? Or are there better methods to obtain the ...
4
votes
2answers
2k views

Proof of the “Radius of Convergence Theorem”

I can't figure out how it is valid to invoke the Absolute Convergence Theorem, whose hypothesis is "Let the power series have radius of convergence R", to establish case c of the Radius of Convergence ...
3
votes
1answer
173 views

Remainder term in Taylor's theorem

I'm trying to understand the remainder in Taylor's theorem from this source: https://proofwiki.org/wiki/Taylor%27s_Theorem/One_Variable/Integral_Version I don't understand the very last parts of ...
3
votes
5answers
427 views

Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
3
votes
3answers
2k views

General term of Taylor Series of $\sin x$ centered at $\pi/4$

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 ...
2
votes
2answers
134 views

Taylor series of $[\log(1-z)]^2 $

I'm having some trouble proving that the Taylor series about the origin of the function $[Log(1-z)]^2$ to be $$\sum_{n=1}^\infty \frac{2H_n}{n+1}z^{n+1}$$ where $$H_n = \sum_{j=1}^n \frac{1}{j}$$ So ...
2
votes
4answers
156 views

Two ways to show that $\sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$

Show that: $\large \sin x -x +\frac {x^3}{3!}-\frac {x^5}{5!}< 0$ on: $0<x<\frac {\pi}2$ I tried to solve it in two ways and got a little stuck: One way is to use Cauchy's MVT, define $f,g$ ...