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
28 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 ...
0
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1answer
16 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 ...
2
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3answers
74 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) = ...
0
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1answer
21 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} ...
2
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4answers
92 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 ...
1
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1answer
26 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 + ...
0
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1answer
21 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 ...
0
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0answers
41 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 ...
0
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2answers
54 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 $e^{x^2+x}$: ...
0
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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 ...
3
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0answers
170 views
+100

Is there a way to write this recurrence relation in faster-to-program manner?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
0
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1answer
92 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 $f(\frac{1}{n}) = 0$ ...
0
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2answers
30 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 ...
0
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1answer
28 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 ...
3
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4answers
89 views

How to evaluate this limit using Taylor expansions?

I am trying to evaluate this limit: $\lim_{x \to 0} \dfrac{(\sin x -x)(\cos x -1)}{x(e^x-1)^4}$ I know that I need to use Taylor expansions for $\sin x -x$, $\cos x -1$ and $e^x-1$. I also realise ...
2
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4answers
157 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 ...
0
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0answers
15 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} ...
5
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1answer
299 views

How to compute the values of this function ? ( Fabius function )

How to compute the values of this function ? ( Fabius function ) It is said not to be analytic but $C^\infty$ everywhere. But I do not even know how to compute its values. Im confused. Here is the ...
0
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1answer
611 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 ...
7
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2answers
102 views

How can I get f(x) from its Maclaurin series?

I know how to get a Maclaurin series when $f(x)$ is given. I have to find $\sum_{n=0}^{\infty}\frac{f^{(k)}(0)}{k!}x^k$. But how can I get $f(x)$ from its Taylor series? The problem is $$f(x) = ...
0
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1answer
20 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 ...
0
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0answers
5 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 $[-1,2]$. ...
2
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1answer
30 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}= ...
0
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1answer
20 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 ...
1
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1answer
20 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 ...
8
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4answers
697 views

Derivation of the Boltzmann factor in statistical mechanics

I have seen similar derivation of the Boltzmann factor many times before, (http://micro.stanford.edu/~caiwei/me334/Chap8_Canonical_Ensemble_v04.pdf , just for example), which I think is incomplete. ...
0
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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 ...
0
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0answers
33 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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0answers
27 views

Why is it that when n ≥ 1 the series is $\le$ 1/4 [closed]

So how is the series $\sum_{n=1}^\infty \frac{1^2 * 3^2 * 5^2 ... (2n-1)^2}{2^2 * 4^2 * 6^2 ... (2n)^2}$ < 1/4 for n $\ge$ 1 is it because the series is divergent outside of the interval of ...
0
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0answers
17 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 ...
1
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1answer
27 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 ...
1
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2answers
17 views

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 ...
0
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1answer
15 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 - ...
0
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0answers
14 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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0answers
28 views

4th derivative of $1 - 9x + 16x^2 - 25x^3 + \dots$ [closed]

Fined the 4th derivative of $f$ at $x=0$ given that the MacLaurin series of $f$ is $f = 1 - 9x + 16x^2 - 25x^3 + \dots$.
0
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2answers
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 ...
1
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4answers
107 views

What does it mean that a taylor series generated for a function f(x) doesnt converge to f(x)?

If a some function f(x) is continous and has derivatives of all orders on some interval I, and assuming that f(x) can be expressed as a power series on I. And now you generate a taylor series for ...
2
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1answer
29 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 ...
0
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0answers
37 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. ...
2
votes
2answers
96 views

Series expansion for integral including error function

$\DeclareMathOperator{\erfc}{erfc} \DeclareMathOperator{\Ei}{Ei} $ What is the series expansion of $f$ for small $q$? \begin{align} U(q) &= q e^{q^2}\erfc q\\ I(q,q') &= ...
4
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4answers
110 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!} + ...
2
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2answers
41 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 $x = 0$?
1
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1answer
48 views

Estimating $\int_0^{\sqrt 2 / 2} \sin (x^2) dx$ with Taylor Series

I seem to be having trouble with part of this question (Reference: Apostol Volume 1, Section 7.8, Question 8). The full question states: (a) If $0 \leq x\leq \frac{1}{2}$, show that $\sin x = x - ...
0
votes
0answers
21 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 ...
0
votes
0answers
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 ...
0
votes
1answer
52 views

Taylor expansion of $\frac1{\|x-y\|}$

Let $0\neq y\in \mathbb{R^3}$ define a function $f$ on $\mathbb{R^3}$ as $$ f(x) = \frac1{\| x-y\|} $$ What are derivatives of $f$ in zero? Or equivalently, what is the Taylor series of $f$ at ...
0
votes
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 ...
1
vote
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 ...
-2
votes
0answers
12 views

Determining second order Taylor polynomial of function f at two points

Can someone help me determine the second order Taylor approximation of the function 1/(1+x+(y)2) near (0,0) and near (1,0)? This is in the context of directional and partial derivatives.
0
votes
0answers
31 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 ...