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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Factorization of Taylor series.

I know that for a (finite) polynomial $P(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_0$ whose zeros are $x_1, x_2, \ldots, x_n$, then we can factorize it as $$P(x) = a_n(x - x_1)(x - x_2) \cdots (x -...
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35 views

Can we Relate Radius of Convergence of Taylor Series and Asymptotic Rate of Growth?

I still need to be disabused of the belief that there is some simple connection between the finiteness of the radius of convergence and the asymptotic rate of growth. 1. Can we develop any ...
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1answer
20 views

Is Every (Real) Analytic Function (with Non-Degenerate MacLaurin Series) Asymptotically Greater Than any Polynomial?

Question: Given a function $f: \mathbb{R} \to \mathbb{R}$ such that the MacLaurin series exists and equals the function for every $x \in \mathbb{R}$, and such that for all $n \ge n_0$, $n_0$ some ...
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36 views

Taylor series Lagrange Remainder explanation

So, given a Taylor series: $$f(x)=f(x_0)+f'(x_0)(x-x_0)+f''(x_0)\frac{(x-x_0)^2}{2!}+\cdot\cdot\cdot+f^{(n)}(x_0)\frac{(x-x_0)^n}{n!}+R_n$$ The error $R_n$ is given by: $$R_n=\frac{f^{(n+1)}(\xi)}{(n+...
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1answer
19 views

Could someone please confirm my answer this Maclaurin series??

Find three nonzero terms of the Maclaurin series of the function $f(x)={3/5} tan5x/x$ Using the maclaurin series i found them to be.. $3/5+x^2/25+2x^4/25$ Is this correct? If not what is the ...
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2answers
27 views

Could some confirm my answer for this limit using taylor series?

$\lim_{x→0}$ $\dfrac{x^2}{x\sqrt{1+x} −\ln(1+x)}= ?$ I got $-2$. Is this correct if not what is the answer so i can find out where i went wrong. Thanks in advance
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47 views

Find a function f so that Taylor expansion is always accurate to this degree

Find a function $f$ from R to N such that with $T$ be the Taylor expansion of $\sin(x)$ around $0$. $ | \sin (x) - T_{f(x)}x$| $\leq 1$ The hint is to use $n! \leq 3 \sqrt{n} {(\frac{n}{e})}^n$
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Taylor series question help!

This question is on a past paper for my exam but no model solutions have been provided and I'm worried I'm doing completely the wrong thing, Consider two functions represented by Taylor (MacLaurin) ...
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22 views

Taylor expansion in proof of weak maximum principle

Picture below is part of proof of weak maximum principle. On the red line ,I don't know how to use the Taylor expansion to get $-u''(x_0) \le 0$. I think the Taylor expansion of $u(x)$ at $x_0$ is $$ ...
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19 views

Taylor series roots at infinity

I started thinking about this after this MathSE thread. Take a sequence of Taylor polynomials $f_n$ that converge to $f$. Does $f_n$ always have a growing number or roots in $\mathbb{C}$ which grow ...
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3answers
40 views

Complex analysis: Using Taylor expansion to show $|c_n| ≤ \frac{1}{r^n}\sup_{z∈C_r(0)}|f(z)|$

Consider the function $f$ is defined through the power series $$f(z) := c_0 + \sum_{n=1}^\infty c_nz^n$$ and assume that the series on the right has a radius of convergence $R > 0$. Show that $$|...
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1answer
32 views

Limit calculate using Maclaurin series

I need help to calculate this limit using Maclaurin series: $\lim_{x\to \infty}((x^3-x^2+\frac{2}{x})e^{\frac{1}{x}}-\sqrt{x^3+x^6})$ I don't know from where to start. I think I need to to write ...
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1answer
23 views

Polynomial approximation of a limit

I am supposed to find the Taylor polynomial $P_2(x;1)$ for the exponent function $f(x)=e^x$ and use it in conjunction with Taylor's theorem to evaluate the following limit: $$\lim_{x\rightarrow1} \...
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1answer
31 views

$\lim_{x \to 0}\frac{(x^2 \times 2^x \times (\log 2)^2) - (2^x - 1)^2}{(2^x - 1)^2(x^2 \times \log 2)} = ?$

$$\lim_{x \to 0}\frac{(x^2 \times 2^x \times (\log 2)^2) - (2^x - 1)^2}{(2^x - 1)^2(x^2 \times \log 2)}$$ I tried this by using the Taylor series $2^x = 1 + x\log 2 + \frac{x^2}{2!}(\log2)^2 + \dots$....
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1answer
27 views

Mclaurin series

I need to use Mclaurin series in order to show that for every $x\in$ $(0, 1)$ $\sin(x)>\ln(1+x)$ I don't know from where to start, I think I should define $f(x)=\sin(x)-\ln(1+x)$ and then to ...
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4answers
107 views

prove: $\lim\limits_{x \rightarrow a} \frac{f^2(x)- g^2(x)}{(f(x) -f(a))^2} = 1$

$f(x)$ and $g(x)$ both differentiable twice at $x = a$ and we know that $f''(a) =g''(a)+f(a)$, $f(a) = g(a) = f'(a) = g'(a) \not = 0$ (we don't know if $f(x)$ and $g(x)$ are differentiable anywhere ...
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1answer
39 views

Integrate $\frac{\sin x^3}{x^3}$ as a power series

Today, I tried to do this by taking the MacLaurin of Sin to 4 terms, putting in $x^3$ in place of $x$, multiplying the terms by $x^{-3}$, and integrating. I came out with a sum that had $x^{6n+1}$ as ...
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108 views

prove: if $f(\frac{1}{n}) = 0$ for all $n \in \mathbb{N}$ then $f(x)=0$ for all $x \in \mathbb{R}$

$f(x)$ is infinitely differentiable and $\exists L \in \mathbb{R}$ such that $|f^{(n)}(x)| \le L$ for any $n \in \mathbb{N}$. I need to prove that given the information above: if $f(\frac{1}{n}) = 0$ ...
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2answers
42 views

The value of $\lim_{n\rightarrow\infty}[(n+1)\int_{0}^{1}x^{n}$ $\ln(1+x)$ $dx]$

I evaluated it as $\lim_{n\rightarrow\infty}[x^{n+1}ln(1+x)]_{0}^{1}-\int_{0}^{1}x^{n+1}(1+x)^{-1} dx$ , which comes as $\ln (2) - \lim_{n\rightarrow\infty}\int_{0}^{1}x^{n+1}(1-x+x^{2}-x^{3}\ldots)dx ...
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45 views

Approximating functions using Taylor polynomials

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide any explanations. I literally have no idea how to ...
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1answer
32 views

How can I evaluate the limit of this function using series?

Limit as x approaches 0 of $lim_{x\rightarrow 0}\frac{1-cosx}{1+x-e^x}$. I substituted in the Taylor series of $cosx$ and $e^x$ into the function, but it's still in $\frac{0}{0}$ form, and I don't ...
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3answers
87 views

Show that $\sin(x) > \ln(x+1)$ for any $x \in (0,1)$

Show that $\sin(x) > \ln(x+1)$ when $x \in (0,1)$. I'm expected to use the maclaurin series (taylor series when a=0) So if i understand it correctly I need to show that: $$\sin(x) = \lim\...
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16 views

Proof relating to the remainder term in Taylor's theorem

I'm asked to show that $\left\lvert R_n(x) \right\rvert \leq \frac{\left\lvert x \right\rvert ^n}{n!}\sup \left\lvert f^{(n)}(t)\right\rvert$ where $$R_n(x)=\frac{1}{(n-1)!}\int_0^x (x-t)^{n-1} f^{(n)}...
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4answers
178 views

Show $\int_0^1 \frac{\log(1-x)}{x}dx=-\frac{\pi^2}{6}$

It's claimed that $$\int_0^1 \frac{\log(1-x)}{x}dx=-\frac{\pi^2}{6}$$ by first expanding $\frac{\log(1-x)}{x}$ into a power series and then doing term-by-term integration. I want to justify this by ...
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6 views

Methods for bounding the remainder of taylor expansion for $e^x$

When I was reviewing taylor series, I ran into trouble on the following problem. Problem: Write down a Taylor polynomial to compute $e^x$ within $10^{-3}$ of error on the interval $[-1,2]$. ...
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1answer
22 views

How to find the upper bound of an error by Taylor polynomial approximation

I'm struggling about finding a way to find the upper bound of the error of Taylor polynomial approximation. I will explain better using a solved example I found... $f: ]-3;+\infty[ \rightarrow \...
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1answer
34 views

Is there any known way to sum a subserie (square indices) of geometric series?

I was interested in the following sum. Although im not sure there exist any known way to sum this...it seems rather difficult. Can we sum for $0<r<1$ $$\sum_{k=0}^{\infty}r^{k^2}= 1+r+r^4+r^{9}+...
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1answer
24 views

Taylor Maclaurin series

Can someone explain to me how this equals? I'm taking a calculus III course at the moment, and I'm doing Taylor and Maclaurin series at the moment, and this is the last step of a problem, but i don't ...
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2answers
60 views

What's wrong with my Taylor -Maclaurin- Series? $e^{x^2+x}$

Here's what I have: We know: $$e^x = 1 + x + \frac{1}{2!}x^2+\frac{1}{3!}x^3 +\frac{1}{4!}x^4$$ Now I can calculate the Taylor Series for $e^{x^2+x}$: $$1+u+u^2+\frac{1}{2!}(x^2+x)^2+\frac{1}{3!}(...
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1answer
38 views

I like to know if there is approximated expression of something

I like to know if there is approximated expression of below things. $$e^{-x}\sum^{9}_{k=0}\frac{1}{k!}x^{k}$$ $x$ is not always small. I know behind part is Maclaurin series but summation range is ...
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35 views

Regarding a proof in Tu's 'Introduction to manifolds'

While reading Tu's differential geometry book, I came across a theorem which makes the following claim: Let $f$ be a $\mathcal{C}^\infty$ function on an open subset $U\in \mathbb{R}^n$, let $p\in U$, ...
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1answer
27 views

Taylor expand $\ln(x) - \ln(1-y)$ around$(\ln(x'),\ln(y'))$

Can I taylor expand $$\ln(x) - \ln(1-y)$$ around $(\ln(x),\ln(y'))$ such that I get $$ \ln(x') - \ln(1-y') + \frac{\partial (\ln(x) -\ln(1-y))}{\partial \ln(x')} (\ln(x) - \ln(x')) + \frac{\partial (\...
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20 views

Big 'O' Notation - Taylor Series

Q) Use the Taylor Series Expansion to show the first derivative f '(x) can be approximated by $$-(3f(x) -2f(x+h) - f(x-h) / h ) $$ What is the precision? Now I found after using the Taylor ...
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1answer
30 views

Question Regarding Proof of Taylor Remainder Theorem in Tu's “An Introduction to Manifolds”

The statement: Let $f$ be a $C^{\infty}$ function on an open set $U\subseteq \mathbb{R}^n$ which is star shaped with respect to a point $p=(p^1,...,p^n) \in U$. Then there are functions $g_1(x),...,...
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Can the original function be derived from its $k^{th}$ order Taylor polynomial?

Coming from a statistics background, I'll provide an example related to fitting a model to an analysis dataset. Let's suppose I suspect the relationship between the mean value of the outcome variable (...
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1answer
19 views

First-order Taylor series expansion

I have a first-order equation that is supposed to be solved using the Frobenius method. I am having some difficulty since the equation is not equal to zero. I would appreciate any help. y' + (1 - x^...
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20 views

If $f$ and $h$ are differentiable in $a$ and $h'(a)=f'(a)$. Then $g$ is differentiable and $g'(a)=f'(a).$

Let $f,g,h:X\to\mathbb{R}$ such that $f(x)\leq g(x)\leq h(x)$ for all $x\in X$. If $f$ and $h$ are differentiable in $a$ and $h(a)=f(a)$. Then $g$ is differentiable and $g'(a)=f'(a).$ How can I can ...
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45 views

A fast converging limit for $\ln x$ (or why $\ln 2 \approx \sqrt{\sqrt{42}-6})$

I've read a lot about approximating logarithms recently, and apparently it's not easy. It can be done by Taylor series (slow convergence), by continued fractions (also slow) and also by some limits. ...
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1answer
33 views

Order of remainder term in Taylor series approximation

I'm having trouble verifying a bound on the remainder term of a Taylor series approximation. I have a $C^\infty$ function $f$ of compact support. Using the two-term Taylor series for $f$ centered at $...
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113 views

Finding out a limit using Taylor series.

So the limit is the following: $$\lim_{x \to 0}{\frac{x^2-\frac{x^6}{2}-x^2 \cos (x^2)}{\sin (x^{10})}}$$ Expansions for $\sin(x)$ and $\cos(x)$ are given: $$\sin x = x-\frac{x^3}{3!} + \frac{x^5}{...
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2answers
42 views

Is $f(x)=\sum_{n\geq 1}\frac{(-x)^n}{n^2+1}$ convex at $x=0$?

Let $\sum_{n=1}^{\infty}\frac{(−1)^n}{ n^2+1} x^n$ be the Taylor series of $f(x)$ about $0$. Then, is it that, $f(x)$ is concave up at $x = 0$?
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26 views

Hermite Expansion of Probability Density Function

While reading this paper by Ait-Sahalia I got stuck with a formula which is quite important. Nevertheless it is not derived explicitly by the author. I resume here the main steps, it is quite a long ...
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19 views

Taylor expansion of the inverse of a function

The Taylor expansion of a function is given by $$ f(x) \approx x^{(1)} + x^{(2)} + x^{(3)} $$ From this we can establish $$ \frac{1}{f(x)} \approx \frac{1}{x^{(1)} + x^{(2)} + x^{(3)}} $$ Is there a ...
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15 views

Approximating a quadratic term in the constraint set as 2nd order Taylor expansion

I have an optimization problem in the following form: $$\min_{x,y} f(x)+g(y)$$ $$s.t.$$ $$Ax+h(y)=0$$ where $h(y)$ is a quadratic in $y$. Instead of solving this problem directly, $h(y)$ is ...
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1answer
31 views

Can someone draw a plot for this function?

Can someone draw a plot for this function? $ f(x) = \begin{cases} \sum_{i=2}^{x}\left(\frac{\prod_{k=1}^{i-1}\left(2k-1\right)\,\cdot\,-\left(-\frac{1}{2}\right)^{i}}{i!}\right) + \frac{3}{2} & x ...
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2answers
57 views

If $|f(x)| \leq 1$ and $|f''(x)| \leq 1$, show $|f'(x)|\leq 2$

Given $f : \mathbb{R} \to \mathbb{R}$, such that $f'(x)$ and $f''(x)$ exist for all $x \in \mathbb{R}$ and for $x \in [0,2]$, the inequalities $|f''(x)| \leq 1$ and $|f(x)| \leq 1$ hold, I am asked to ...
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1answer
41 views

Does multiplying Taylor series by an integer change the interval of validity.

If I have a Taylor series for example, $\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \ldots, \qquad \text{valid for $-1<x<1$} $ and I multiply the series by some integer, let's say $5$, in ...
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1answer
20 views

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$.

If $f(x)=\sum_{0}^{\infty}b_n(x-5)^n$ for all $x$, write a formula for $b_8$. Now I know that $b_n=\dfrac{f^{(n)}(5)}{n!}$. I have tried various things but I think there is something wrong with my ...
0
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2answers
16 views

Calculating Taylor series of complex function

I'm going through a past exam paper and found a question I can't do. The question is to write down the Taylor expansion of $\frac{z^2}{z-2}, z \in C$ \ {2}, on the disc $|z| < 2$ I've been ...
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1answer
39 views

Why does $\frac{\frac12 x+\frac18x^2+O(x^3)}{\frac12x-\frac18x^2+O(x^3)}=1+\frac12x+O(x^2)$?

I was reading the solution to a limit through Taylor expansion but did not understand this passage: $$g(x)=\frac{1-\sqrt{1-x}}{\sqrt{1+x}-1}=\frac{\frac12 x+\frac18x^2+O(x^3)}{\frac12x-\frac18x^2+O(...