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.

learn more… | top users | synonyms (2)

0
votes
1answer
33 views

Real Analysis: Bounds for derivatives using Taylor's Theorem

Suppose that $f''$ exists on [0,1] and that $f(0)=0=f(1)$. Suppose also that $|f''(x)|\leq K$ for $x\in(0,1)$. Prove that $|f'(1/2)|\leq K/4$ and that $|f'(x)|\leq k/2$ for $x\in(0,1)$. I'm trying to ...
0
votes
0answers
30 views

Finding a Maclaurin series

I have a question here; suppose $f(x)= x^2\sin(x^3)$ By using the Maclaurin series for sine, find the Maclaurin series for $f$ I understand how to obtain the Maclaurin series for $f$ using the ...
0
votes
2answers
55 views

Analytic functions equal to all orders in a point are equal on the open interval

Let $A\subset \mathbb{R}$ be open. To make everything clear, my definition of analytic function here is: A function $\psi : A\to \mathbb{R}$ of class $C^\infty$ is said to be analytic if for each ...
0
votes
1answer
54 views
1
vote
1answer
69 views

Asymptotics and little-o notation

I always have issues dealing with asymptotic notation... I am trying to verify the following step: $$\left(1-\frac{t^2}{2n} + o(1/n)\right)^n \to e^{-t^2/2}.$$ To change this into ...
1
vote
1answer
40 views

How to show the existence of an entire function

I have been working on this problem for quite sometime. For part (i), I obtained the Taylor series for $4\sin(z) - \sin(4z)$. At $z = -\pi$, the Taylor series is: $4\sum_{n=0}^{n} \frac{(z + ...
0
votes
2answers
46 views

Compact Form of the Taylor Series

Determine the Taylor Series $\frac{1}{\sqrt{1-x}}$ at $x=0$ I ended up with this: $1 + \frac{1}{2}x+\frac{3}{4}x^2\frac{1}{2!}+\frac{15}{8}x^3\frac{1}{3!}+\frac{105}{16}x^4\frac{1}{4!}$ I am ...
4
votes
1answer
84 views

Limitations of fractional derivative approximation with Taylor series

I was playing around with the concept of fraction derivatives, and came across some base functions for which it is defined, namely power and exponential functions $$ \left(\frac{d}{dt}\right)^\alpha ...
2
votes
2answers
96 views

taylor expansion of $\sinh(x)$

I would like to find taylor expansion of $sh(x)$ My thoughts indeed, note that : $\sinh(x)=\dfrac{e^{x}-e^{-x}}{2}$ then \begin{align} \sinh(x)&=\frac{e^x-e^{-x}}{2} \\ ...
3
votes
1answer
59 views

Different convergence radius for different power series of the same function.

i was playing around with $$\frac{1}{x^2+x+1}$$ I got 3 different series's: $$\sum_{n=1}^\infty (\frac{x}{(x+1)^2})^n*(\frac{1}{x}) $$ which converges when $|\frac{x}{(x+1)^2}|<1$ the second ...
2
votes
2answers
52 views

To show that the limit of the sequence $\sum\limits_{k=1}^n \frac{n}{n^2+k^2}$ is $\frac{\pi}{4}$

Show that $$\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{n}{n^2+k^2} = \frac{\pi}{4}.$$ I am familiar with Taylor series and Fourier series of the standard functions. I tried to compare with those ...
0
votes
2answers
37 views

Taylor series for df

So I understand if I have f(x) under a taylor expansion I can write the terms up to order 2 terms as: f(x)= f(a) + f'(a)(x-a) + [f''(a)*(x-a)^2]/2! +... so I would imagine df(x)/dx = f'(a) + ...
0
votes
0answers
26 views

$\ln(1+x)\underset{x\to 0}{=} \sum_{k=1}^{n}(-1)^{k+1}\frac{x^{k}}{k}+o(x^{n})$

I would like to see the setps behind that implication $$ \frac{1}{1+x}\underset{x\to 0}{=} \sum\limits_{k=0}^{n-1}(-1)^{k}x^{k}+o(x^{n-1}) \implies \ln(1+x)\underset{x\to 0}{=} ...
0
votes
1answer
47 views

Leading order Taylor Series Represention of the following function

I am given with this function $$f=\frac{1}{\sqrt{1+af_1(x)+bf_2(x)}},$$ where $$f_1=(1+x^2)^\nu,$$ and $$f_2=x^2(1+x^2)^{\nu-1},$$ where $\nu$ is a rational constant. I would want my $f$ to be of the ...
0
votes
2answers
48 views

Taylor expansion for vectors

$$F(x,y)= (x_2-x_1^2) (x_2-2x_1^2)= 2 x_1^4+x_2^2-3x_1^2x_2$$ Where $x^*=[x_1 \ \ x_2]' = [0 \ \ 0]'$ I want to show Taylor expansion of the function for third degree. What I did is that; ...
1
vote
1answer
23 views

Reducing terms in the series expansion of a function of two variables

I have a function $f(x, y)$. This function is such that \begin{align} f(0, y)=a\\ f(x, 0)=a, \end{align} where $a$ is a constant. From this, a particular mathematician concludes: Thus if we ...
4
votes
1answer
84 views

On the binomial series $(1+\frac{1}{8n})^{1/2}$, where $n$ is an even perfect number

Since $\sqrt{1+8n}=\sqrt{8n}\sqrt{1+\frac{1}{8n}}$, and $\frac{1}{8n}<1$ when $n>1$ is an integer, then we can express the real number $\sqrt{1+\frac{1}{8n}}$ by its binomial series. This series ...
0
votes
2answers
52 views

How to prove $\frac{x}{e^x-1}=1-\frac{1}{2}x+\frac{1}{12}x^2+o(x^2),(x\to0)$ using Taylor's Formula?

$$\frac{x}{e^x-1}=1-\frac{1}{2}x+\frac{1}{12}x^2+o(x^2),(x\to0)$$ I have attempted to expand the multinomial $e^x-1$ by using Taylor's Formula, and I got this: ...
1
vote
1answer
32 views

approximating $(1-e^{-x})^2$ near $x=0$ with $x^2$ via Taylor expansion

I would like to show that $(1-e^{-x} )^2$ is approximated well near $x=0$ with $x^2$ via Taylor expansion but can't quite seem to complete the job. I know that by expanding the exponential into its ...
0
votes
1answer
21 views

How to find the degree- n term in the Maclaurin polynomial of $f(x)=\ln(1+x)$?

How to find the degree- n term in the Maclaurin polynomial of $f(x)=\ln(1+x)$? My Thoughts: The nth term is obviously: $$\frac{f^{(n)}(0)}{n!}x^n$$ But I am stuck here, how do I find the ...
0
votes
0answers
12 views

Finding the Interval of Convergence for the Sum/Difference of two Power Series

The question: Find the Taylor Series for $$f(x) = \frac{1}{x^2-3x-18}$$ at x = 1. Find the interval of convergence. My work: $$\frac{1}{x^2-3x-18} = ...
0
votes
1answer
184 views

Cauchy Product of two Taylor Series

I'm probably being a bit stupid here but I've been assigned this question and don't really know where to go with it. Compute the first 5 terms of the Cauchy product of the Taylor Series for ...
0
votes
1answer
48 views

Series expansion at infinity

I am trying to find to generalize the limit that involves all rational functions such as $\sum_{n=0}^{l}\frac{{a}_{n}{x}^{n}}{{b}_{n}{x}^{n}}$. I believe the best way of generalizing all of them is ...
0
votes
0answers
12 views

Bounding the error for $e^{x+5y}$ taylor polynomial expansion

The exercise asks me to prove: $$|e^{x+5y}-P_1(x,y)|< \frac{3}{2}(x+5y)^2$$ when $x+5y<1$ I don't understand what's the exercise suggesting but I tried this: $e^{x+5y} - P_1(x,y)$ is just ...
4
votes
1answer
202 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 $x+b$: ...
1
vote
1answer
29 views

Taylor polynomial of degree 2 of $e^{x^2+x}$

I want to find the Taylor polynomial of degree 2 of $e^{x^2+x}$ and this is what the answer should be: $$e^{x^2+x} = e^{x^2}e^{x} = (1 + x^2 + O(x^4)) (1 + x + \cfrac{x^2}{2} + O(x^3)) = 1 + x + ...
2
votes
2answers
76 views

Taylor series of a convolution

The derivation below is from Probability Theory: The Logic Of Science By E. T. Jaynes, chapter 7 "The Central Gaussian, Or Normal, Distribution", p.706 The Landon derivation. Text available online: ...
0
votes
1answer
21 views

Taylor expansion of difference of functions

Is the taylor expansion of the difference of functions (more specifically the difference of the same function at different points) simply the difference of the taylor expansions? Since that may be ...
1
vote
2answers
38 views

Estimating error using Taylor Polynomial

I have searched and read quite a bit on this subject but I can't get this last bit straight. Reading the other answers did not help me unfortunately for me. Anyway the problem: Suppose I have the ...
0
votes
0answers
63 views

Is e^(-1/x) a flat function at x = 0?

Taylor series of e^(-1/x) at x = 0 shows that it is flat function on x= 0. But in every text on flat function I see the example of the function e^(-1/x^2) and not e^(-1/x). I am starting to think that ...
1
vote
3answers
45 views

Prove that the series $ \sum_{n=0}^\infty \frac{(-1)^n \cdot x^{2n}}{(2n)!} $ represents $ \cos x $ for all values of $ x $

guys. The question is as stated in the title: prove that the series $ \sum_{n=0}^\infty \frac{(-1)^n \cdot x^{2n}}{(2n)!} $ represents $\cos x $ for all values of $ x $ My doubt is quite ...
0
votes
1answer
23 views

How large need n to be to ensure that Taylor polynomial around x=0 gives a value of sin(pi) which has an error of less than 0.001?

I've found different methods to calculate $n$, but all include that I test it for several $n$. Is it possible to make a general formula that gives me the answer without having to test it, or do I need ...
0
votes
1answer
23 views

expressing $p , p(p+1) , p(p+1)(p+2)$ as a series

I'm working on arithmetical analysis and more specifically on finite differences. I want to create a series consisting of the following terms : $$f(x_{0} + ph) = f_{0}+ p\Delta f_{0}+ ...
1
vote
1answer
51 views

Taylor Series Expansion of $ f(x) = \sqrt{x} $ around $ a = 4 $

guys. Here's the exercise: find a series representation for the function $ f(x) = \sqrt{x} $ around $ a = 4 $ and find it's radius of convergence. My doubt is on the first part: I can't seem to find ...
0
votes
1answer
19 views

Write the indicated case of taylor's formula

I have this problem: "Write the indicated case of Taylor's formula for the given function. What is the Lagrange remainder in each case? $f(x) = \ln{x}$ $a = 1, n = 6$ " That's the information I ...
0
votes
0answers
26 views

Is there a way to separate this function?

let $f(\mathbf r_1,\mathbf r_2) = \frac{1}{|\mathbf r_1 - \mathbf r_2|^2 + a^2}$. Is there a method to represent this as (a series of) separated functions in the form?: $f = \sum \limits _i ...
0
votes
0answers
34 views

How to derive analog of power rule for other forms of the derivative?

Introduction We'll be dealing with multiple forms of calculus here. So we'll use $\operatorname{L_d}(f(x))$ to refer to the additive derivative, $\cfrac{df}{dx}$, $\operatorname{P_d}(f(x))$ to refer ...
1
vote
2answers
47 views

What is the trick to Taylor expand this function to 4th order?

The function is $ u(x,y)= -x-y-xyu^3$, and I want to Taylor expand $u(x,y)$ around (0,0) in powers of x and y to 4th order. To first order, I differentiated implicitly, and the expansion is: ...
0
votes
0answers
26 views

remainder term error in maclaurin polynomial

Consider function f(x)=$\frac{1}{1-x}$, find the remainder term Rn(Z) of a function of x and n. I now know that $f^{(n)}(x)=\frac{n!}{(1-x)^{n+1}} $ and that ...
0
votes
1answer
33 views

Is the remainder of first-order Taylor expansion still continuously differentiable?

Let $f: {\mathbb R}^n \to {\mathbb R}^n$ be a continuously differentiable function. Then, we can rewrite its first-order Taylor expansion at $x \in {\mathbb R}^n$ for $h \in {\mathbb R}^n$ that ...
0
votes
1answer
64 views

Find $n$ such that $\tan 1$ and its Taylor series up to $n$ agree to 1000 decimal places

I know the Taylor series for $\tan x$ is, $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$ I am trying to find a value for $n$ such that $|\tan 1 ...
1
vote
1answer
42 views

Find the Mclaurin series of $f(x)=(1+x)^\alpha$ then find the radius of the convergence by ratio test.

Find the Mclaurin series of $f(x)=(1+x)^\alpha$ then find the radius of the convergence by ratio test. I'm told that i should not assume n is an integer. I think to find the Mclaurin series i ...
0
votes
0answers
13 views

Expanding a Function into a Maclaurin's Series

"Given that $y^3=e^xcosx$, show that $3y^2\frac{dy}{dx}-y^3=-e^xsinx$. By further differentiation of this result, or otherwise, find the Maclaurin series for y, up to and including the term in ...
0
votes
1answer
18 views

Limit is infinite or finite?

The given function is $${{Log\ z}\over {z-1}}=1- {1\over 2}(z-1)+ {1\over 3} (z-1)^2-{1\over 4}(z-1)^3+.... $$ Then it is said that the function tends to $+\infty$ as $z$ tends to $0$ . But ...
2
votes
0answers
105 views

How to use Taylor series to get $e^x\geq1+x$

I know that from $$e^x=\sum_{i=0}^\infty \left(\frac{x^i}{i!}\right)$$ we can get the inequality $e^x\ge1+x$. But how?
1
vote
2answers
54 views

How do I evaluate this without using taylor expansion :$\lim_{x \to \infty}x^2\log(\frac {x+1}{x})-x\ $?

How do I evaluate this without using Taylor expansion? $$\lim_{x \to \infty}x^2\log\left(\frac {x+1}{x}\right)-x$$ Note: I used Taylor expansion at $z=0$ and I have got $\frac{-1}{2}$ Thank ...
0
votes
0answers
40 views

Radius of convergence of a Taylor-series

I came across the following question. Let $c \in \mathbb{R}$ and let $f: \mathbb{R} \to \mathbb{R}$ be defined by: $$ f(x) = \frac{1+ c x^2}{1+ x^2}$$ Let $c \neq 1$. Determine for $a=0$ the ...
0
votes
0answers
17 views

Functions on a connected subspace

For one of my courses I came across the following question. Let $U$ be a connected supspace of $\mathbb{C}$ and let $f: U \to \mathbb{C}$ and $g: U \to \mathbb{C}$ be complex analytic (eg complex ...
0
votes
0answers
23 views

Can I have a proof of the Taylor Series of $f$?

My textbook states the following: Taylor Expansion: $f(x)=\frac{f(a)}{0!}+\frac{f^{\prime}(a)(x-a)}{1!}+\frac{f^{\prime\prime}(a)(x-a)^2}{2!}...$ I actually have used this a lot in my own personal ...
0
votes
0answers
78 views

How to prove the following Taylor expansion for twice differentiable functions

I want to prove the following: Let $f(x)$ be a twice differentiable function. Then, $$\begin{array}{l} \exists t \in \left[0,1 \right ] \; s.t., \\ f\left(y \right ) = f\left(x \right ) + \left(y-x ...