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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320 views

Stirling's Formula - Comparison Test Method

The following question concerns the convergence of Stirling's Approximation for $n!$ I have $r_n = \frac{\sqrt{n}}{n!}(\frac{n}{e})^n$. I have expressed ...
2
votes
2answers
167 views

Taylor expansion with change of variables question.

Find the Taylor polynomial of order 3 of $$f(x,y) = (x - 1)^{2} + \sin(\pi y) + x \ln(y)$$ based at $(x,y) = (2,1)$. So I'm really lazy and don't want to take the derivative of that, so let ...
3
votes
2answers
64 views

Prove that $\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$ where $|R_8(x)|\leq \frac{x^8}{8!}$

Prove that $$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+R_8(x)$$ where $|R_8(x)|\leq \frac{x^8}{8!}$
4
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1answer
88 views

Let $f(x)=e^{-1/x}$ for $x>0$ and $f(x)=0$ for $x\leq 0$. Prove that $f^{(n)}(0)=0$ for all $n\in \Bbb N$.

Let $f(x)=e^{-1/x}$ for $x>0$ and $f(x)=0$ for $x\leq 0$. Prove that $f^{(n)}(0)=0$ for all $n\in \Bbb N$. I'm reading the solution, and I understand how to prove that all derivates must be of the ...
0
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2answers
70 views

Is it true that if the taylor series of $f$ converges it converges to $f$?

Is it true that if the taylor series of $f$ converges it converges to $f$? So if I want to prove that $\lim R_n(x)=0$, could I prove this using the fact that a given taylor series converges ?
3
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1answer
679 views

Find the taylor series for $\cos x$ and indicate why it converges to $\cos x$ for all $x\in \Bbb R$.

Find the taylor series for $\cos x$ and indicate why it converges to $\cos x$ for all $x\in \Bbb R$. I've posted my own proof, I hope it is correct :)
2
votes
0answers
84 views

Series expansion of a series

I would like to perform an asymptotic expansion of the function $$f(x) = \sum_{n=1}^{\infty}\frac{1}{(nx)^2}K_2(n x),$$ where $K_2(x)$ is the modified Bessel function of the second kind, around $x=0$. ...
0
votes
2answers
953 views

Find the Taylor series for $\sinh(x)$ and indicate why it converges to $\sinh(x)$.

Find the Taylor series for $\sinh(x)=\frac 1 2(e^x-e^{-x})$ and indicate why it converges to $\sinh(x)$.
3
votes
2answers
949 views

Proof of the “Radius of Convergence Theorem”

I can't figure out how it is valid to invoke the Absolute Convergence Theorem, whose hypothesis is "Let the power series have radius of convergence R", to establish case c of the Radius of Convergence ...
4
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1answer
92 views

$\frac{\mathrm d^n}{\mathrm d x^n} e^{-\frac {1}{x^2}} = 0$ at $x=0$ [duplicate]

This is an exercise from David Brannan's Mathematical Analysis. I've proved parts (a) - (c) but need help with Part (d). Any guidance appreciated. EDIT I have solved it, by induction using the ...
4
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1answer
111 views

Write down the equation of the tangent plane and compute the Taylor series of the function

Set $f(x,y,z) = x + y + z + x^2 + y^2 + z^2$. Consider the surface $$S = \{f(x,y,z) = 0\} \subset \mathbb{R}^3$$ near the origin $o = (0,0,0) \in S$. Write down the equation of the ...
4
votes
3answers
326 views

Let $g(x)=e^{-1/x^2}$ for $x\not=0$ and $g(0)=0$. Show that $g^{(n)}(0)=0$ for all $n\in\Bbb N$.

Let $g(x)=e^{-1/x^2}$ for $x\not=0$ and $g(0)=0$. Show that $g^{(n)}(0)=0$ for all $n\in\Bbb N$. In the text it is already proven that for the function $f$ with $f(x)=e^{-\frac{1}{x}}$ for $x>0$ ...
2
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1answer
65 views

Taylor series representation of a function.

I'm working on expressing the function $f(x)=\frac{6}{x}$ as a taylor series about $-4$. I've got the general idea, but I'm not quite there yet. I've come up with the equation ...
2
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2answers
54 views

Find the Taylor series of $\frac{1}{x+1} $ at $x=2$

This is what I did: $\begin{align*} f(x)&=&(x+1)^{-1}\\ f'(x)&=&-(x+1)^{-2}\\ f''(x)&=&2(x+1)^{-3}\\ f'''(x)&=&-6(x+1)^{-4}\\ f''''(x)&=&24(x+1)^{-5}\\ ...
2
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4answers
748 views

What exactly are the “higher order terms” (H.O.T.) in Taylor series expansion in multivariable case?

I know the Taylor series expansion in single variable case: $$ f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 + \frac{1}{3!}f^{(3)}(x_0)(x-x_0)^3 + \frac{1}{4!}f^{(4)}(x_0)(x-x_0)^4 ...
2
votes
3answers
352 views

Complex Analysis Taylor Series

So the problem states: "Say f(z) := log z is the principal branch of the logarithm (the primitive of 1/z on the region C(-infinity,0]). Show that the Taylor series of f(z) about $z_0 = -1 + i$ takes ...
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2answers
190 views

Taylor Expansion of the 1/2th Derivative

In trying to solve the problem $\sqrt D f(x)=g(x)$ I tried to expand the derivative as a Taylor series, and have encountered a lot of problems. Is there some reason that this shouldn't be possible? ...
2
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4answers
127 views

Taylor expansion of $x^x-1$ around 1

How do I find Taylor expansion of(around 1): $$f(x)=x^x-1$$ The answer should be: $$(x-1)+(x-1)^2+\frac 12(x-1)^3+\cdots$$ How the answer was obtained?
4
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1answer
799 views

Find Maclaurin series of $(\sin(x^3))^{1/3}$

How do I find Maclaurin series for the function: $$\sqrt[3]{\sin(x^3)}$$ The answer should be: $$ x - \frac {x^7}{18} - \frac {{x}^{13}}{3240} + o(x^{13})$$ I tried: $$\sin x = x - \frac ...
2
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2answers
230 views

Maclaurin series for $\frac{x}{e^x-1}$

Maclaurin series for $$\frac{x}{e^x-1}$$ The answer is $$1-\frac x2 + \frac {x^2}{12} - \frac {x^4}{720} + \cdots$$ How can i get that answer?
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1answer
29 views

Some question about Taylor polynomia

$$f(t) = \frac{100e^t}{10+e^t} \text{for all } t \in \mathbb{R}$$ (1) show that $$\frac{d}{dt}f(t)=f(t) - \frac{1}{100} (f(t)^2)$$ (2) waht is the second derivative $f^{(2)}(t)$ in term of $f(t)$? ...
1
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1answer
85 views

How to compute tangent of a discretized curve

I have a discretized curve defined by a 2D matrix $M$ where $M(i,j)=1$ means the point $(i,j)$ is on the curve. For each of these points, I want to calculate its tangent vector by fitting a polynomial ...
1
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1answer
75 views

Radius of convergence of a sum of two series

Assume that the radius of convergence for $\sum_{k=0}^{\infty} c_k x^k$ is $11$ and that the radius of convergence for $\sum_{k=0}^{\infty}d_k x^k$ is $13$. Determine the radius of convergence for ...
2
votes
1answer
57 views

Bounding a continuously differentiable function using Taylor given the function is bounded by the norm of x

Suppose $0 < r < 1$ and that $f \colon B_1(0) \to \mathbb R$ is continuously differentiable. If there is an $\alpha > 0$ s.t. $|f(x)<\Vert x\Vert^\alpha$ for all $x \in B_r(0)$, prove ...
4
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1answer
110 views

A sort of “Taylor expansion” of a power series

I have the following question. Suppose $$f(x):=\sum_{i=0}^{\infty}c_ix^i$$ is a power series that converges for $|x|<1 + \epsilon$, for some $\epsilon >0$, where $x\in\mathbb{C}$. I can then ...
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2answers
44 views

$R_{n}\left ( x \right )$ for $ f\left ( x \right) = \cos\left ( 2\cdot x \right ) $?

How can I define $R_{n}\left ( x \right )$ for $$ f\left ( x \right) = \cos\left ( 2\cdot x \right ) $$ I found taylor expansion for $cos2x$.What should I do after that? My problem is I dont know if ...
6
votes
4answers
897 views

Using Taylor's Theorem to show that $\ln(1 + x^2) \leq x^2$

Can we show that if $\operatorname{abs}(x) \lt 1$, then $$\ln(1+x^2) \leq x^2\;,$$ using Taylor's Theorem? I am thinking of expanding it about $x=0$ but I got something like $$f(x) = -x^2 + ...
0
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1answer
369 views

Laurent Series and Taylor Series

I am trying to find the Laurent series of $\dfrac{1}{(1+x)^3}$; would this be the same as finding the Maclaurin series for the same function?
1
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1answer
257 views

limit of the error in approximating definite integral with midpoint rule

I want to calculate $\lim_{n \rightarrow \infty} n^2 |\int_{[0,1]}f(x)-I_n(x)|$ where $I_n$ is the integral approximation by midpoint rule: $I_n=\frac{1}{n}\sum_{k=1}^nf(c_k)$ and $c_k$ is the point ...
1
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1answer
62 views

Taylor polynomial

I need your help to solve this question. I tried something, but i can't finish my proof. Let $f(x)$ be a differentiable function in $(0, \infty )$, so that $|f'(x)|$ is bounded there. Prove that ...
1
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1answer
362 views

Taylor series for logarithm converges towards logarithm

Is there a way to show that the Taylor series around 0 of $f(x) = \ln(1-x)$ converges towards $f$ on the interval $(-1,1)$, just by considering the remainder from the Taylor polynomial? I'm having a ...
0
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0answers
115 views

using Taylor series to prove an inequality

Prove that if $p^T▽f(x_k)<0$, then $f(x_k+εp)<f(x_k)$ for $ε>0$ sufficiently small. I think we can expand $f(x_k+εp)$ in a Taylor series about the point $x_k$ and look at $f(x_k+εp)-f(x_k)$, ...
0
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1answer
65 views

Taylor s inequality

Apply Taylor´s inequality to derive the quadratic Taylor approximation of $e^x$ at $x=0$. Could anyone help me out? I tried looking up the definition but I am not sure what is meant by "at $x$ is ...
2
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1answer
159 views

Bounding approximation error for Taylor polynomial

I've got this problem: Let $f(x) = e^x$. If we aproximate $f(x)$ by $P_4(x)$ in $x_0 = 0$ at $(-r, r)$, find $r \gt 0$ so that the error in the approximation is $\lt 10^{-5}$ What I did is: 1) ...
1
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1answer
98 views

Exact expansion of functions

Prove that for any twice differentiable function $f: {R}^n \to R$, $f(y) = f(x) + \nabla f(x)^T (y-x)+ \frac{1}{2} (y-x)^T \nabla^2f(z)(y-x) $, for some $z$ on the line segment $[x, y]$. Note that ...
0
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1answer
91 views

Maclaurin vs Taylor and their geometrical difference

In this topic i learned how to approximate a function with a high degree polynomial and how to derive the Maclaurin series: $$ f (x) = P_n(x) = f(0)+{f'(0)\over 1!}x+{f''(0)\over ...
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1answer
298 views

Taylor expansion of an integral

I am interested in the Taylor series expansion around $t=0$ of the following expression: $$I(t)=\int_{0}^{\infty}e^{-x^2}\log\left(e^{-(x-t)^2}+e^{-(x+t)^2}\right)dx$$ Normally, I would proceed by ...
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1answer
372 views

Taylor / Maclaurin series expansion origin. [closed]

Soo we all know Taylor series expansion formula for expansion around expansion point $A(a,f(a))$: $$f(x) \approx \underbrace{f(a)}_{1st~term} + \underbrace{\frac{f'(a)\, (x-a)}{1!}}_{2nd~term}+ ...
0
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1answer
635 views

Finding the 4th order Taylor expansion of $g(t)= t^3 + 2t^2 + 2t + 1$

Given the function $$g(t) = t^3 + 2t^2 + 2t + 1$$ I would like to find the 4th order expansion of $g(t)$ at $t=t_1$. So far, I have performed the differentiation of $g$, up to $g'''(t)$ w.r.t. $t$, ...
5
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1answer
208 views

Application for mean value theorem

$f(x)$ is three-times differentiable on $[a,b]$, how to show that there is $\varepsilon\in(a,b)$ such that $$f(b)=f(a)+\cfrac{1}{2}(b-a)[f'(a)+f'(b)]-\cfrac{1}{12}(b-a)^3f'''(\varepsilon)$$
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1answer
64 views

Expansion of $x^{-1/2}$ at $0$

Regard the function $f(x) = x^{-1/2}$ on the non-negative real line. The point $z=0$ is 'distinguished' because it is the boundary of the domain of $f$, and because $f$ has a pole there. It seems ...
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5answers
124 views

Proof that $2.82<\pi<3.19$

Using taylor expansion of $\cos$ function. What I have is $$1-\frac{x^2}{2}+\cdots-\frac{x^{4n-2}}{(4n-2)!}<\cos(x)<1-\frac{x^2}{2}+\cdots+\frac{x^{4n}}{(4n)!}$$ How would I proceed from here? ...
2
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2answers
87 views

Find $\displaystyle\lim_{x \rightarrow 0} \frac{e^{\sin x} - e^x}{\sin^3 2x}$

I have to find $\displaystyle\lim_{x \rightarrow 0} \frac{e^{\sin x} - e^x}{\sin^3 2x}$ using Taylor polynomials. Here's what I've done so far: $e^x = 1 + x + \frac{1}{2} x^2 + \frac{1}{6} x^3 + ...
3
votes
1answer
182 views

Complex Taylor series and Bernoulli numbers

Let: $f(z)=\frac{z}{e^z-1}$ if $z\ne0$, and $f(z)=1$ if $z=0$. Please help to prove that $\sum_{k=0}^{n-1}\binom n kf^{(k)}(0)=0$ for any $n>1$ and $f^{(2n+1)}(0)=0$ for any natrual $n$.
2
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2answers
578 views

Taylor expansion for $\sqrt{x+2}$

I'm enrolled in Coursera's calculus with a single variable and am trying to solve one of the homework problems. In lecture, it was stated that to expand $\sqrt x$ about $x=a$, you would have: ...
1
vote
3answers
1k views

How to expand $\tan x$ in Taylor order to $o(x^6)$

I try to expand $\tan x$ in Taylor order to $o(x^6)$, but searching of all 6 derivative in zero (ex. $\tan'(0), \tan''(0)$ and e.t.c.) is very difficult and slow method. Is there another way to ...
5
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1answer
169 views

Taylor's Theorem Application Question, $f(x)$ smooth and $f(0)=0$ implies $f(x)/x$ smooth.

I am wondering the following fact, and I believe I know the answer, but I am not sure why. If $f(x)$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}$, if $f(0)=0$, is it true that $f(x)/x$ is ...
3
votes
1answer
71 views

What's incorrect with this Taylor series derivation?

Let's \begin{align} f(T)= &f(0)+ \int_0^T f' (t)dt\\ f(T)=&(0)+f' (T)T-\int_0^T f'' (t)tdt\\ f(T)=&(0)+f' (T)t-f'' (T) \frac{T^2}{2}+\int_0^Tf''' (t) \frac{t^2}{2} dt\\ f(T)=&f(0)+f' ...
0
votes
1answer
311 views

When finding upper bound for error, can $\xi$ be different from $x$?

The question is to find $P_3(x)$ for $ f(x) = (x-1) \; \ln x $ about $x_0 = 1$ and find the upper bound on the error for $P_3(0.5)$ used to estimate $f(0.5)$. I got $$ f^{(4)}(x)= \frac{2}{x^3} + ...
4
votes
5answers
122 views

$\lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $ solution?

I recently took an math exam where I had this limit to solve $$ \lim_{x \to 0} \frac {(x^2-\sin x^2) }{ (e^ {x^2}+ e^ {-x^2} -2)} $$ and I tought I did it right, since I proceeded like this: 1st I ...