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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Use a calculator to compute the error $|e^x-T_2(x)|$ at $x=1.1$

I don't believe i have learned to solve for the error. Any help would be greatly appreciated. I have computed $T_2$ at $x=0.8$ $$T_2=e^.8+e^.8(x-.8)+e^.8/2(x-.8)^2 $$
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2answers
97 views

Computing $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$.

Compute $\int_{0}^{1\over 2}{\ln(1+x)\over x}dx$ with a precision (Accuracy? Error? What is the formal expression?) of 0.01. Attempt: First of all: $\ln(x+1)=\sum_{k=1}^{\infty}{(-1)^{k-1}x^k\over ...
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0answers
19 views

Condition on the limit of the upper incomplete gamma function

I am trying to find a lower bound on $q$ such that $$\Gamma(2p,qt)=\int_{qt}^{\infty}{x^{2p-1}e^{-x}\,dx}>0$$ for all $t>0,p>0$, and $q<0$. At first, I tried expanding $\Gamma(2p,qt)$ ...
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1answer
67 views

How to use Chebyshev Polynomials to approximate sin(x) and cos(x) within the interval [−π,π]? [on hold]

I have approximated sin(x) and cos (x) using the Taylor Series (Maclaurin Series) with the following results How can I use Chebyshev Polynomials to approximate sin(x) and cos(x) within the ...
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0answers
63 views

Validity of approximating a difference equation with a differential equation

Consider the following two equations: $$ \begin{cases} A_k(n+1)-A_k(n)=\beta \displaystyle \frac{ A_{k-1}(n)- A_k(n)}{\alpha+ n} + \delta_{k, \beta} \\ \\ \displaystyle ...
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2answers
33 views

$\ln $ and Taylor Series Expansion (what went wrong)

Edited Problem I'm trying to express $\ln{(1-(\frac{N}{K})^{\frac{1}{4}})}$ in terms of $\ln N$, where $K$ is a constant and $1 \leq N \leq K$. This also implies $\frac{N}{K} \leq 1$. Anyone ...
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3answers
29 views

Proof:Taylor expansion of inverse trigonometric functions

I find it quite difficult to remember the Taylor expansion of inverse trigonometric functions.Actually in school we have been just taught the series (for finding limits in calculus without teaching us ...
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1answer
39 views

Comparing Two Multilinear Polynomials based on Multivariable Taylor Expansion

Given two linear functions $f(x)$ and $g(x)$ defined on real values, let's say that I want to show that $f(x) > g(x)$ for all real $x > t > 0$. According to the order-1 Taylor expansion at ...
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1answer
22 views

Why is this true - easy question concerning asymptotics of exponential

Suppose $\lambda > 0$ is constant as $t \searrow 0$. In my lecture notes it is written that $\left(1+\sum_{k=1}^{\infty} \frac{(-\lambda t)^k}{k!}\right) \lambda t = \lambda t + o(t)$ and ...
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0answers
65 views

A derivation of the Euler-Maclaurin formula?

The generating function for the Bernoulli numbers $B_n$ is $$\frac{x}{e^x-1}=\sum_{n=0}^\infty\frac{B_n}{n!}x^n$$ The sum of an infinite geometric series is $$\frac{1}{1-x}=\sum_{k=0}^\infty x^k$$ ...
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2answers
60 views

A Taylor series expansion of $e^{ix}$

In Probability Theory by Athreya and Lahiri, they give a very elegant proof of Central Limit Theorem (The Lindeberg one) wherein they use a lemma: For $x \in \mathbb{R}$ and $r \geq 1$, ...
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0answers
20 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 $x+b$: ...
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0answers
26 views

Continuity of $f^{(n-1)}$ in Taylor's Theorem with Mean-value remainder

I refer to Rudin's proof of Taylor's Theorem with the Mean-value form of the remainder. I'm not sure if I'm understanding the proof correctly. Why must $f^{(n-1)}$ be continuous on $[a,b]$? I ...
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1answer
67 views

Coefficient of operator and how to do it

This question stems from this $$ \frac{1}{x+z}- \frac{1}{x} = \sum_{k=0}^\infty \frac{z^k}{k!}\frac{d^k}{dx^k}[\frac{1}{x}] $$ Now, i need to find the Bell Polynomial of $\frac{1}{x}$, $$ ...
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2answers
35 views

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence.

Find Taylor expansion of $f(x)=\ln{1-x^2\over 1+x^2}$, and then find radius of convergence. My problem here is the Taylor series. Computing the few first derivative is possible, but I can't seem to ...
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1answer
86 views

If $f(0)=0$ and $f''\ge 0$, then $f(a+b)\ge f(a)+f(b)$

Given $\ f$ so $\ f''(x) \ge 0$ for every $\ x \ge 0$, also $\ f(0)=0$. Trying to show that if $\ a,b \ge 0 \Rightarrow f(a+b) \ge f(a) + f(b)$ Using Taylor I used $\ f(0)=0$ and got ...
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1answer
31 views

Given that $f(1) = f'(1) = 1$, use Taylor polynomials to show that $\lvert f(x) - x \rvert \leq A(x - 1)^2$

Given that $\ f$ has continuous second derivatives in$\ [0,2]$ and $\ f(1)=f'(1)=1$, I'm trying to prove that for every $\ x \in [0,2]$ exists an A so that: $$ |f(x)-x| \le A(x-1)^2 $$ The second ...
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4answers
1k views

What's the idea behind the Taylor series?

I understand that they are viewed as approximations, but was that Taylor's original hope? Assuming that a function can be written as a power series seems to me to be a wild assumption, without some ...
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0answers
42 views

How to determine Taylor series expansion of function $f(x) = \frac{\cos(x)}{x}$ about $a=1$?

Given function is $f(x) = \frac{\cos(x)}{x}.$ $y = x - a , y = x - 1$. $x = y+1 , f(y) = \frac{\cos(y+1)}{y+1}$ How to get Taylor series expansion about $1$ of this function? If it was needed to ...
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3answers
94 views

Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$ [duplicate]

Determine the Taylor series expansion of function $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}}$. $f(x) = \frac{\arcsin(x)}{\sqrt{1-x^2}} = \arcsin(x)\frac{1}{\sqrt{1-x^2}}$ It is known: (1.) ...
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1answer
39 views

Prove that for every $x$ in the area of $0$ exists: $\ln(1+x)=\sum_{1}^{\infty}\frac{(-1)^{n+1}x^n}{n}$

I need to prove 2 things: Prove that for every $x$ in a neighbourhood of $0$ exists: $\ln(1+x)=\sum_{1}^{\infty}\frac{(-1)^{n+1}x^n}{n}.$ What I did is that I calculated the derivatives of ...
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5answers
2k views

Why do un-integrable funcitons 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
65 views

Determine the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$

I need to calculate the sum $\sum_{n=0}^{\infty}{(-1)^n}\frac{2^{2n-1}}{(2n)!}$ . It seems very "similar" to Taylor expansion of functions arcsin(x) and its derivative for x = -2. It is known: ...
4
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0answers
66 views

On applying Whitney's extension theorem to suitable closed sets

Whitney's extension theorem states that if $D \subset \mathbb{R}^n$ is closed and $f: D \to \mathbb{R}$ is $C^k$ in some sense to be specified below, then $f$ can be extended to $\mathbb{R}^n$ so that ...
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1answer
26 views

When is it appropriate to neglect all terms after the first non-zero term of a Taylor expansion series?

Suppose I am interested in the Taylor expansion series of a Cosine function at the neighbourhood of a=0. In computing the series from n=0 to n = infinity, when would it be appropriate to neglect all ...
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1answer
18 views

Cropping off the Taylor Series

We know that the Taylor series is for expansion of any function, but for digitization we need to crop off some parts? How can we determine upto which derivative should we consider.. I am mainly ...
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0answers
12 views

Using Taylor series of function f1(x) in Taylor series of function f2(x) defined in open discs D1 and D2 when D2 lies inside D1

We have two open discs, D1 and D2, whose centres are C1 and C2 respectively. The Taylor series of function f1(x) is defined in open disc D1 while the Taylor series of function f2(x) is defined in open ...
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5answers
66 views

$\lim_{x \to 0} \cfrac{e^{2x} - \ln(1-x) - \sin(x)}{\cos(x)-1}$ using Taylor Expansions

As a preface- a very similar question is here: Using Taylor expansion to find $\lim_{x \rightarrow 0} \frac{\exp(2x)-\ln(1-x)-\sin(x)}{\cos(x)-1}$ But, my actual question is, when we substitute the ...
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1answer
25 views

error term for double integral approximation by midpoint rule

I found following statement in the book that I'm reading: Using Taylor series expansions it is easy to prove that: $$ \left|h^2\cdot ...
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1answer
56 views

How to solve this limit using laurent series?

$$\lim_{x\to\infty}\left(\left(\frac{x^2+5}{x+5}\right)^{3.7}+\left(\frac{x^3+5}{x+5}\right)^{1.6}\right)^{20/37}-\left(\left(x-5\right)^{3.7}+(x^2-5x+25)^{1.6}\right)^{20/37}=60$$ It is possible to ...
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1answer
14 views

Taylor expansion of $f$ in stability analysis of 2-step Adams-Bashforth method

Given the two-step Adams-Bashforth method $$ u_{n+1} = u_n + \tfrac{h}{2}(3f_n - f_{n-1}) $$ find its order. Some notation: $t_n = t_0 + nh$ is the $n$-th node and $y_n = y(t_n)$; $f_n$ ...
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2answers
77 views

Expand the Taylor series for the following mind-boggling expression at $x = 0$

Mind-boggling expression is: $$f(x) = \frac{x^{15}}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)}$$ I started by using the quotient rule and expanding the denominator terms in the hopes of finding some pattern ...
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1answer
33 views

Does an inequality between first-order Taylor approximations imply the same for the functions?

Assume that $f$ and $h$ are functions from $\mathbf{R}^n$ to $\mathbf{R}^1$ and continuously differentiable. Also assume that $f(z)=h(z)$ at some point $z \in \mathbf{R}^n$. Could we then show that ...
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1answer
340 views

Taylor Series Theorem

So I see the argument presented in taylor series, that $$\sum c_n (x-a)^n = \sum \frac{f^{(n)}(a)}{n!} (x-a)^n$$ or $c_n = f^{(n)}(a)/n!$ if $x=a$ the question is, since the above only holds when ...
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1answer
31 views

Proving that a function is increasing

I have this problem Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a function, such that its Taylor series convergers to function $f$ everywhere. For every derivative of the function $f$ we have that ...
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1answer
28 views

Computing first three non-zero terms of a Taylor series

I have a function $F(t)=\int_0^t \sqrt{1-x^8} dx.$ I have to find the first three non-zero terms of a Taylor series of $F$ around the point $a=0.$ Since I want the Taylor series I started with the ...
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1answer
84 views

Taylor series approximation of function under norm

I am reading this paper. At page number $4$, term $||Au - f||^{2}$ is approximated by taylor series approximation around $u^{k}$. The resulting approximations are $$\|Au - f\|^{2} \approx ...
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1answer
622 views

Taylor Series Expansion for $\tan x$

I'm trying to determine the Taylor series expansion for $\tan x$: I know that the $n$th derivative of the expansion must be the same as the $n$th derivative of the function. Please help, I have no ...
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0answers
13 views

Signs and value of higher order terms in the Taylor expansion of a strongly convex function

Say that I have a function $f: S \rightarrow \mathbb{R}^+$ such that: $S \subset \mathbb{R}^n$ is a closed convex set such as $S=[-10,10]^n$ $f$ is continuous and infinitely differentiable at all ...
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2answers
33 views

Extending a bounded holomorphic function past its boundary

Suppose I have a bounded holomorphic function on the unit disc, centred at the origin. Can I always extend this beyond the origin to say a disc of radius $1 + \epsilon$ for some $\epsilon > 0$? My ...
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1answer
400 views

How many iterations of Taylor series for n correct decimal digits

I'm using Taylor series to estimate trigonometric functions. So I need to know exactly how many iterations of Taylor series (say for sine) are needed for n decimal digits precision? (I'm writing a ...
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0answers
27 views

Steady state response approximation of a linear differential equation using Taylor polynomial

After thinking out how to convert a non-homogeneous linear differential equation, with a polynomial input, to a homogeneous linear differential equation in general for this question I started playing ...
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3answers
56 views

How to expand $\sqrt{x^6+1}$ using Maclaurin's series

The expansion would be $\sum_{n=0}^\infty$$\frac{1}{2}\choose n $$x^{6n}$ How to evaluate binomial coefficient with rational numbers? If $\frac{1}{2}\choose n $=$2n\choose n $$\times ...
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0answers
21 views

Does small perturbation in the denominator explode the expectation of a ratio of two random variables?

The puzzling thing I am facing is Suppose we have two random variables $X$ and $R$ such that $E(X^{-1}R)=1$. Now let $\tilde{X}=X+\mathcal{E}$ where $\mathcal{E}=X\epsilon$ and $\epsilon \sim ...
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1answer
16 views

Index confusion in Tu's treatment of Taylor's Theorem with remainder in “An Introduction to Manifolds”

In Tu's book, specifically the section on "Taylor's Theorem with remainder", there appears to be a changing of the meaning of some subscripts which isn't noted. The theorem states that if $f$ is a ...
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1answer
35 views

Taylor expansion at discontinuous point

a) Find the Maclaurin expansion of the following function: $$f(x)=\int\limits_0^x \frac{1-e^{-t^3}}{t^2} \mathrm{d}t$$ end b) evaluate the $ \displaystyle \lim_{x \to 0^{+}} f^{(29)}\, (x) $ The ...
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1answer
34 views

approximation of $\pi$ by $\arctan$

Determinate the order n of the Maclaurin polynomial for $f(x)=4tan^{-1}x$ so that the remainader term $|R_{n}(1)|<0.000005$. Here $R_{n}(1)=\frac{f^{(n+1)}(c)}{(n+1)!}$ for some c between 0 and 1 ...
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1answer
13 views

Use the remainder estimation theorem to find the maximum value of error?

My answer for part (bi) was $\frac{x}{2}$ - $\frac{x^2}{4}$ My attempt for part b(ii) was to find $\frac{g^{'''}(z)}{3!}$($x^3$) = $\frac{8}{3!(2+2z)^3}$ where z ∈ [0, $\frac{1}{2}$]. To find the ...
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5answers
89 views

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$.

Find $\lim_\limits{x\to 1^-}{\ln x\cdot \ln(1-x)}$. I can't even start because I don't really know what $x\to 1^-$ means. If you know what it means it would really help me. I would as well if you ...
0
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1answer
18 views

The Maclaurin series and taylors theorem for $\sinh(2x)$

I am currently studying for an exam next week but am struggling to the second part of this question. I have figure out the Maclaurin series for $\sinh(2x)$, however am unsure how to estimate the ...