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.

Filter by
Sorted by
Tagged with
8
votes
4answers
3k 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: $\sum_{n=0}^{m}n2^n = 2(m2^m - 2^m +1)$ ...
23
votes
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^...
120
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)? ...
5
votes
3answers
245 views

Find the power series of $f(x)=\frac{1}{x^2+x+1}$

I want to find the power series of $$f(x)=\frac{1}{x^2+x+1}$$ How can I prove the following? $$f(x)=\frac{2}{\sqrt{3}} \sum_{n=0}^{\infty} \mathrm{sin}\frac{2\pi(n+1)}{3} x^n \,\,\,\, |x|<1$$ In ...
20
votes
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 ...
6
votes
4answers
15k 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 ...
25
votes
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 ...
15
votes
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 ...
4
votes
4answers
763 views

Show $\ln2 = \sum_\limits{n=1}^\infty\frac1{n2^n}$

Problem: Show that $$\ln2 = \sum_\limits{n=1}^\infty\frac1{n2^n}.$$ My progress: The problem before this one had me find the Taylor series for $\ln(1-x)$ which was $$-\sum\limits_{n=1}^\infty \frac{...
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\...
6
votes
3answers
807 views

How to see $\cos x \leq \exp(-x^2/2)$ on $x \in [0,\pi/2]$?

Can anyone help me with the above inequality? I tried looking at the series expansion and I guess the answer indeed lies there, but I fail to see it. Thanks
57
votes
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: ...
64
votes
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: ...
22
votes
6answers
17k views

An intuitive explanation of the Taylor expansion

Could you provide a geometric explanation of the Taylor series expansion?
14
votes
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 ...
7
votes
2answers
16k views

Multiplying Taylor series and composition

I have two questions: A. I know the taylor series for $\arctan(x)$ and for $e^x$. How do I find the series for $\arctan(x)\cdot e^x$ ? B. Say I want to find the series for $\arctan(g(x))$, do I just ...
14
votes
2answers
944 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
1answer
4k 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. Isn't ...
48
votes
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 ...
12
votes
1answer
2k views

Exponential of powers of the derivative operator

A translation operator The Taylor series of a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{(\partial_x^nf)(a)}{n!}(x-a)^n$$ where $\partial_x$ is the derivative operator. Expanding about $x+b$: $$...
8
votes
4answers
273 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 $$\...
5
votes
4answers
9k views

Taylor expansion of $\arccos(1-x)$ around $x=0$ to two terms

Doing a normal Taylor expansion of $\arccos(1-x)$ around $x=0$ to two terms by taking derivatives doesn't work because of division by zero. I've put this into wolfram alpha: http://www.wolframalpha....
4
votes
2answers
27k views

Taylor series of $\sqrt{1+x}$ using sigma notation

I want help in writing Taylor series of $\sqrt{1+x}$ using sigma notation I got till $1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\ldots$ and so on. But I don't know what will come in ...
4
votes
2answers
2k views

Taylor's polynomial uniqueness proof - why are these limits inferable?

So I'm viewing a short proof on the uniqueness of Taylor polynomials. Uniqueness of Taylor polynomial: Let $f:]a,b,[ \rightarrow \mathbb{R}$ $n$ times continuously differentiable and $x_0 \in ]...
30
votes
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}^\...
5
votes
2answers
1k 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 ...
6
votes
4answers
603 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 ...
7
votes
4answers
11k views

Maclaurin series for $\frac{x}{e^x-1}$

Maclaurin series for $$\frac{x}{e^x-1}$$ The answer is $$1-\frac x2 + \frac {x^2}{12} - \frac {x^4}{720} + \cdots$$ How can i get that answer?
5
votes
6answers
183 views

Maclaurin Expansion for $e^{e^{z}}$ at $z=0$

I need to find terms up to degree $5$ of $e^{e^{z}}$ at $z=0$. I tried letting $\omega = e^{z} \approx 1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\cdots$, and then substituting these first few terms ...
5
votes
3answers
213 views

computing the series $\sum_{n=1}^\infty \frac{1}{n^2 2^n}$

$$\sum_{n=1}^\infty \frac{1}{n^2 2^n}$$ I am new in series thus I tried a pair of methods to compute but I couldn't
3
votes
5answers
12k views

how to prove that $\ln(1+x)< x$ [duplicate]

I want to prove that: $\ln(x+1)< x$. My idea is to define: $f(x) = \ln(x+1) - x$, so: $f'(x) = \dfrac1{1+x} - 1 = \dfrac{-x}{1+x} < 0, \text{ for }x >0$. Which leads to $f(x)<f(0)$, ...
12
votes
0answers
274 views

If $X_i∼fλ$, $Z∼\mathcal N(0,I_d)$ and $Y=X+\ell d^{-α}Z$ with $α<1/2$, then $\liminf_{d→∞}\text E\left[1∧\prod_{i=1}^d\frac{f(Y_i)}{f(X_i)}\right]=0$

Let $f\in C^3(\mathbb R)$ be positive $g:=\ln f$ $d\in\mathbb N$, $$p_d(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and $\lambda^d$ denote the Lebesgue measure on $\mathcal B(\mathbb R^...
1
vote
1answer
68 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) = -\...
2
votes
1answer
192 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 $\dfrac{1/3}{1-z}+\dfrac{1/3}{2+z}$]. ...
1
vote
1answer
277 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: $R_{n+...
-2
votes
2answers
117 views

Use taylor series to arrive at the expression f'(x)=1/h[-3*f(x)/2+2f(x+h)-f(x+2h)/2]

I'm not really sure how to go about this.. any help is appreciated.
40
votes
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\...
23
votes
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^{...
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 ...
11
votes
1answer
1k 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) = \...
10
votes
7answers
31k views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
10
votes
1answer
258 views

What uniquely characterizes the germ of a smooth function?

Let $X$ be the set of all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which are infinitely differentiable at $0$. Let us define an equivalence relation $\sim$ on $X$ by saying that $f\sim g$ if ...
8
votes
1answer
437 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 my ...
8
votes
3answers
819 views

Why does this infinite series equal one? $\sum_{k=1}^\infty \binom{2k}{k} \frac{1}{4^k(k+1)}=1$

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?
7
votes
1answer
262 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
2answers
144 views

Relationship Between Sine as a Series and Sine in Triangles

I'm taking a single variable calculus course and sine series is defined as $$ \sum_{k=0}^\infty (-1)^k \frac{x^{2k+1}}{(2k + 1)!}. $$ Sine is also defined as opposite length over hypotenuse length ...
1
vote
1answer
156 views

If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$

Prove: if $f''(x_0)$ exists then $\lim\limits_{x \rightarrow x_0} \dfrac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$. I'm not exactly sure how Taylor's theorem fits into all this, but I found the ...
11
votes
1answer
403 views

How to prove that: $19.999<e^\pi-\pi<20$?

I would like to know how to prove $$e^\pi-\pi\sim 20.$$ More precisely, I want to show by using only mathematical tools that, $$19.999<e^\pi-\pi<20$$ I have checked with online ...
9
votes
3answers
347 views

Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$

Prove the following inequality: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ This should be solved using Taylor series. I tried expanding the left to the 5th degree and the right site to the ...
6
votes
3answers
2k 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?