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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3answers
26 views

show equality - binomial formula, taylor?

I am trying to show that this is true using the binomial formula or some taylor expansion: $\frac{1}{(1+\epsilon \sum\limits_{n=0}^\infty Z_n(t) \epsilon^n)^2} = 1 - 2Z_0\epsilon + ...
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1answer
22 views

Numerical analysis: what is the error term for the rule…?

The question goes: derive the error term for the rule $phi$ to approximate the third derivative of f(a). I have attached a screenshot I understand how to take the Taylor series in the hint, but the ...
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2answers
67 views

Estimating $\int_0^1f$ for an unknown Lipschitz $f$ to within 0.0001

A friend of mine has a Lipschitz function $f\colon [0,1]\to\mathbb R$ satisfying (Some more characters, and yet a few more...) $$|f(a)-f(b)| \le 5 |a-b| \qquad\text{for all }a,b\in[0,1],$$ ...
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4answers
3k 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 $I$? ...
4
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1answer
38 views

Maclaurin Series nth Derivative

Find $f^{(2016)}(0)$ if $f(x)=\sin(x^2)$. From the Maclaurin series, $$\sin(x^2)=\sum_{n=0}^\infty\frac{(-1)^nx^{4n+2}}{(2n+1)!}$$ Comparing coefficient, ...
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1answer
13 views

Radius of convergence work check

Original question Find radius of convergence of the Maclaurin series for $f(x)=(4-x)^{-0.5}$ Attemp at solution $$f(x)=\frac{1}{2}\sum_{n=0}^{\infty}\frac{(2n)!(x^n)}{2^{4n}(n!)^2}$$ ratio test: ...
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2answers
67 views

Quick Method to Calculate the Maclaurin Series of $\frac{1}{\sqrt{\cos{x}}} $

I am supposed to calculate the maclaurin series for $\frac{1}{\sqrt{\cos{x}}} $ but I can't seem to figure out an efficient way to go about doing this.
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0answers
58 views

Taylor series for multivalued complex functions (and their use in combinatorics)

As far as I know, it is considered to be a "fact" that by the Generalized Binomial Theorem, the complex function $\sqrt{1 + z}$ has the following Taylor expansion at $z = 0$: $$\sqrt{1 + z} = \sum_{n ...
11
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5answers
2k views

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 ...
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1answer
30 views

Matrix series convergence

Suppose we have the Maclaurin series of a function $f$, and it converges in a radius $R$. Then suppose we define a matrix argument to the function in a similar manner to the exponential definition of ...
2
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0answers
21 views

Taylor's Theorem for a function whose domain is $\mathbb R^n$

In my text book, Taylor theorem: Let $f:A\rightarrow\mathbb{R}$ be of class $C^{r}$ for $A\subset\mathbb{R}^{n}$ an open set. Let $x,y\in A$ , and suppose that the segment joining $x$ ...
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0answers
44 views

Taylor's formula and its quadratic term

I struggle with the following problem: For a function $$f: \mathbb{C} \rightarrow \mathbb{R}~,$$ $f$ attains its maximum for $z_0= e^{i\pi/3}$, $f(z_0)=F_{max}.$ Assume we may use Taylor's theorem ...
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1answer
29 views

Multivariable Taylor's Theorem [duplicate]

I chanced upon this lemma when studying differential geometry which seems to depend on Taylor's theorem, but I have never seen it before, could someone explain how the proof works? I am not sure how ...
2
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1answer
45 views

Find the Maclaurin series for $\ln(2-x)$

A little unsure if the result I got makes sense, so I want to ask here to be sure I am not doing something very silly. The Maclaurin series is given by ...
2
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2answers
59 views

find the value of $\int_0^1(C(-y-1)\cdot\sum_{k=1}^{2016}\frac{1}{y+k})dy$

The problem: find the value of $\int_0^1 (C(-y-1)\cdot\sum_{k=1}^{2016}\frac{1}{y+k}) dy$, where $C(\alpha$) is the coefficient of $x^{2016}$ in the Maclaurin series for $(1+x)^\alpha$ What I ...
2
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2answers
49 views

taylor expansion and limit of a series??

$f(x)=\int_0^xtan^{-1}tdt$ what is the taylor expansion about the origin of this function? and how do i use this to get the limit of the series $1-\frac{1}{2}-\frac {1}{3}+\frac {1}{4}+\frac ...
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9answers
14k 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: ...
4
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3answers
111 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
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0answers
19 views

Integral of Taylor for a Generic Function

I can't seem to figure out the solution to this problem: Problem: Find a series for the following equation and give the first 3 terms and the $nth$ term:$$g(x) = \int_3^xf(t)dt$$ The problem also ...
2
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2answers
103 views

Can we calculate $2^k$ using this easy Taylor series?

Trying to calculate $2^k$ by hand for $k\in[0,1]$, it's tempting to use the Taylor expansion of $x^k$ around $x=1$, to get: $$2^k = 1^k + \frac{k (1)^{k-1}}{1!} + \frac{k(k-1) (1)^{k-2}}{2!} + \ldots ...
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6answers
5k 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 ...
3
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0answers
49 views

Function $s(x)=1+\sum_{k=1}^{\infty} \frac{x^k}{k^k}$ - is there any other way to define it?

This series converges for all $x \in (-\infty, \infty)$, thus the function is analytic on the real line and defined by its Taylor series. However, unlike the exponential function, this one is very ...
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0answers
32 views

Asymptotic to $f^{-1}(f ' (x)) $?

Let $tr(n)$ be the triangular numbers and $te(n)$ be the tetrahedral numbers. $$g(x) := \sum \frac{x^n}{n! 2^{tr(n)}}$$ $g'(x) = g(\frac{x}{2}) $ Now consider the analogue $$ f(x) = \sum ...
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1answer
46 views

Error of Taylor Series?

Part of my assignment is to find the third degree Taylor Series of $\tan(x)$ about $\pi/4$ and then estimate the error of this approximation when evaluated at 0.75. Finding the series was easy ...
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1answer
590 views

Taylor expansion, integration by parts, and the integration of dt.

So my notes say, for a continuous function we have $$ \int_a^x f'(t)dt = f(x) - f(a) \tag 1 $$ which I understand. So re-arranging gives. $$ f(x) = f(a) + \int_a^x f'(t)dt \tag 2 $$ or $$ f(x) ...
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1answer
42 views

Other representation of the Lagrange remainder

so far I've only seen this representation of the Lagrange remainder $R_n=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}$ for some $c$ between $x$ and $a$. $(i)$ Now I came across this representation: ...
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0answers
17 views

Differing notation in compact Taylor series for several variables

I'm a second year mathematical physics student. Wikipedia has a compact definition for the Taylor series in several variables: $$T(\textbf{x})=\sum_{\alpha\geq ...
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0answers
20 views

An approximation of denomiator

I am trying to figure out how to make an approximation of $\frac{1}{x^a+y}$ to separate the term $x$ and $y$? I have tried to use Taylor expansion, but it also left the same denominator term? Thanks ...
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2answers
26 views

Exponent of an Exponential Operator

There is a problem in my textbook that asks me to prove the following: For a bounded operator $A$ on a Hilbert space, prove that: $$(e^A)^n = e^{An} $$ for any natural number, $n$. However upon ...
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0answers
38 views

Find the error in the Maclaurin series for $\ln\left(\frac{1+x}{1-x}\right)$.

I have already that the series is, $$\ln\left(\frac{1+x}{1-x}\right)\approx 2\left(x+\frac{x^3}{3}+\frac{x^5}{5}+...+\frac{x^{2n+1}}{2n+1}\right).$$ The remainder is equal to, ...
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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 ...
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2answers
33 views

Expand $f(z)=\frac{1}{z^2(z-i)}$ in 2 different Laurent expansions around $z=i$ and tell where each converges.

My attempt: $$f(z)=\frac{1}{z^2(z-i)}$$ $$\frac{1}{z^2(z-i)}=\frac{Az+B}{z^2}+\frac{C}{(z-i)}$$ Solving for the unknown constants yields $$A=1$$ $$B=i$$ $$C=-1$$ Thus, $$f(z)=\frac{z+i}{z^2} - ...
0
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1answer
16 views

Trouble with integer partition proof

I am reading Keller & Trotter: Applied Combinatorics, pg. 155, and I am having trouble with an intermediate step in a proof. The proof deals with integer partitions: And the part I can't ...
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0answers
7 views

Taylor expansion of Pearson's sample correlation coefficient

I'm trying to find Taylor series expansion of Pearson's sample correlation coefficient and I don't really know how to do it. Any ideas?
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0answers
26 views

Quality of $E(f(X))\approx f(EX)+\frac 1 2 f''(EX)\sigma_X^2$ approximation

For convex $f$, we have Jensen's lower bound $Ef(X)\ge f(EX)$. What conditions do we need to put on $f,X$ so that the second order expansion in the title would be an upper/lower bound/good ...
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1answer
24 views

How to show $\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}$

How to show the below equation ? $$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)} ~~~~~(t\in \mathbb Z^+)$$
0
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0answers
28 views

Help with taylor series summation?

$f(x) = \sqrt{x}$ When you do expansion, you get the following: $$4 + \frac{x-16}{2^3}\times 1! + \frac{(x-16)^2}{2^8}\times 2! + \frac{3(x-16)^3}{2^{13}}\times 3!$$$$+ ...
0
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1answer
46 views

Solving Second-order non-linear ODE, with fractional expansions

I am solving a differential equation related to fluid mechanics, a rigid air bubble rising towards the surface of a liquid. Doing all of the maths, I have come to this differential equation, which I ...
12
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4answers
170 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 ...
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0answers
92 views

Numerical Computation - Taylor series

I'm taking numerical computation course, and I have a problem with this question: Apply Taylor's formula to obtain a power series approximation about $a=0$ to $\sin(\pi x/2)$. Find the remainder, ...
2
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1answer
292 views

The remainder of Taylor (Maclaurin) series of $\cos(x)$

Something is bothering me with the remainder of the Taylor (Maclaurin) series of $\cos(x)$. The formula of $a_n$ is $(-1)^k \frac{x^{2n}}{(2n)!}$. By Leibniz Theorem, $r_n<a_{n+1}$ which is, ...
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1answer
24 views

Implicit Euler local error issue

I'm not sure I get what's going on here, and online resources are not helpful, at least I didn't find any helpful ones. For the problem: $$ \frac{dy}{dt} = f(t, y(t))$$ a numerical solution for ...
2
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0answers
24 views

An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
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1answer
21 views

Alternate (approximate) form for Hypergeometric function 1F1(0.5, 1.5, -x)

I have the following Hypergeometric function of the first kind: $_{1}F_1(\frac{1}{2}, \frac{3}{2}, -x)$ where $x$ is not negative. This function can also be written as the following series: ...
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3answers
51 views

Find the laurent series for $\frac{1}{z(z-2)^2}$ centered at z=2 and specify the region in which it converges.

My attempt: $$\frac{1}{z(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{A}{z}+\frac{B}{z-2}+\frac{C}{(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{(1/4)}{z}+\frac{(-1/4)}{z-2}+\frac{(1/2)}{(z-2)^2}$$ This is ...
0
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1answer
38 views

Taylor Series Expansion of a function?

So, i was studying my Computer Vision lecture notes and i came across this formula which says Say, i have a function $f(x,y,t)$, $x,y$ and $t$ are the varying factors After $t+ \nabla t$, i have ...
2
votes
2answers
761 views

Series expansion of a function at infinity

I know it is possible to expand an expandable function for a real, and for infinite by setting $x=\dfrac1y$ and then expanding for $0$. But my question is, how do we do if the evaluation of the new ...
0
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0answers
23 views

Loglinearize a nonlinear difference equation

There is an exercise in my weekly problem set I have to solve and I am really struggling. The setting is as follows: Suppose you have the following nonlinear difference equation in terms of Π_t and ...
1
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2answers
27 views

Taylor series of $\ln{\sqrt[4]{\frac{x-2}{5-x}}}$ to $o((x-x_0)^n)$ when $x_0 = 3$

Well I have tried to get it as $$f(x) = f(x_0) + \frac{f'(x_0)(x-x_0)}{1!} + \frac{f''(x_0)(x-x_0)^2}{2!} + ... + o((x-x_0)^n)$$ and got wrong results: First: $$f'(x) = \frac{3}{4(x-2)(5-x)}$$ ...
1
vote
1answer
45 views

Computing limits with Taylor series. [duplicate]

I'm posting my whole thought process, but I'd like to ask specifically about whether is my expansion of $\log(1+y)$ into Taylor series good & allowed in this situation? Is there an easier (not ...