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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16 views

Series expansion with quadratic

If i define a function $$F(t)=f􏰀(x(t),y(t))$$ with $$ x(t)=x_0+∆x * t + ∆^2x * t^2$$ $$y(t)=y_0+∆y * t + ∆^2y * t^2$$ $∆$ is the slope $dx/dt$ and $∆^2$ is the 2nd derivative $d^2x/dt^2$ of ...
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1answer
25 views

Maclaurin series of $f(x)=x^3\sin 2x$

I need help finding that maclaurin series for following function. $$f(x)= x^3 \sin2x$$ How do you get to the maclaurin series?
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1answer
32 views

Non-vanishing terms of Maclaurin series for $\log(3-\cos x^2)$

I have to find the first two non-vanishing terms in the Maclaurin series of $$g(x) = \log(3 − \cos(x^2))$$ and that prove $x=0$ is a stationary point. What is a quick way of working out the ...
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2answers
34 views

Find the first two non vanishing maclaurin terms

Find the first two nonvanishing terms in the Maclaurin series of $\sin(x + x^3)$. Suggestion: use the Maclaurin series of $\sin(y)$ and write $y = x + x^3$ Using this result, find ...
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0answers
11 views

Third order piecewise cubic taylor approximation

I'm trying to approximate a function $d = d(v(t)) = d(x(t),y(t))$ by using a quantised integrator, please see QSS3, page 5. Every step, I know either $x, y$ or both. Both are represented by ...
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0answers
149 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
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1answer
35 views

What's the fourth term in the multivariable Taylor expansion?

For a function $f: \Bbb R^n \to R$, the $2$nd order Taylor expansion is: $$f(\mathbf x+\mathbf h) \approx f(\mathbf x)+ Df(\mathbf x) \mathbf h + \frac{1}{2}\mathbf h^T H(f)(\mathbf x) \mathbf h$$ ...
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1answer
36 views

Finding nth derivative of an exponential function and its value at the origin.

I have a function defined as $f(x) = e^{-\frac{1}{x^2}}, $if $ x\ne0$; $0$ if $x =0$. where $f:[0,\infty) \to \mathbb{R}$ I am asked to prove the following: (a) that the nth derivative is of the ...
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1answer
431 views

Why are there two series representations of the natural logarithm?

On the Wikipedia article of the natural logarithm one finds two different series representations for $\ln(x)$: $\ln(x)= (x - 1) - \frac{(x-1) ^ 2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} \cdots$ ...
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0answers
42 views

Taylor series of $f(x) = \arctan(x)$ converges to $\arctan(x)$

I have to find out the Taylor series of $f(x) = \arctan(x)$ and prove that it converges to $f(x)$ for any $x \in (-1, 1) $. So far I determined the Taylor series to $T_f(x) = \sum ...
3
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1answer
33 views

Bounding the error in the finite difference approximation $\frac{-3f(x) + 4f(x+h) - f(x + 2h)}{2h} - f'(x)$

A course problem asks me, assuming that $f$ is $C^3$ on $\mathbb{R}$ (and $f'''$ is bounded and continuous on $\mathbb{R}$), to show that $$\left| \frac{-3f(x) + 4f(x+h) - f(x + 2h)}{2h} - f'(x) ...
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1answer
19 views

Taylor expansion for left&right limit

I must find the left and the right limits of a function. Can I use the Taylor expansion (at $x_0$) in order to evaluate them?
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2answers
56 views

Can I prove $\sum_{n=2}^{\infty }(1-\frac{1}{n!})\ln^n(2)=\frac{2\ln(2)-1}{1-\ln(2)}$ [closed]

Can I prove $$\sum_{n=2}^{\infty }(1-\frac{1}{n!})\ln^n(2)=\frac{2\ln(2)-1}{1-\ln(2)}$$ depending on Taylor series
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4answers
72 views

Find limit of $\frac {1}{x^2}- \frac {1}{\sin^2(x)}$ as x goes to 0

I need to use a taylor expansion to find the limit. I combine the two terms into one, but I get limit of $\dfrac{\sin^2(x)-x^2}{x^2\sin^2(x)}$ as $x$ goes to $0$. I know what the taylor polynomial ...
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2answers
49 views

What can one conclude for a poynomial with the property that p(x)=p(ix)?

From a theorem in the theory of cyclotomic polynomials I deduced that a special polynomial $p(x)$ of even degree $n$ has the property $$p(x)=p(ix)$$ with $i$ being the complex unit. What can one ...
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0answers
9 views

Remainder of the taylor expansion as a function of the dimension of the space

Given a function $f(x):\mathbb R^n \rightarrow \mathbb R$ under certain conditions it can be approximated with the Taylor expansion: $f(x) = p(x) + r$ where $p$ is a polynomial and $r$ is the ...
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1answer
41 views

Finding the Accuracy of a Taylor Polynomial for the Approximation $f(x) \approx T_{n}(x)$

Let $$ f(x) = \sin(x), \quad a = \frac{\pi}{6}, \quad n = 4, \quad 0 \leq x \leq \frac{\pi}{3} $$ Find a fourth degree ($n=4$) Taylor polynomial for $f$. $$ T_{4}(x) = ...
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2answers
25 views

Non integral exponent for taylor expansion using sage

This is my function var('h,r') f=r^2*arccos((r-h)/r)-(r-h)*sqrt(2*r*h-h^2) taylor(f,h,0,3) Result: ...
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1answer
59 views

estimation of $\pi$ and $e$ by using the Taylor series of $\cos x$

how can one show, that $3<\pi<3.2$, $2.7<e<3$ by just knowing, the estimation of $\cos(x)$, namely: $$1-x^2/2+x^4/24-x^6/720\le\cos(x)$$ ? If I substitute Pi/2 into this estimation, I ...
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0answers
25 views

Solution for a function using Taylor series

How shall I evaluate $0.7^{0.7}$ using the first five terms of the Taylor series for $\ln(1+x)$ and $e^x$?
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1answer
56 views

What does it mean by: Taylor series of $f(1+x)$ converges to $f(1+x)$?

I have $f(1)=0$ and $f'(x)=1/x$. Consider Taylor series of $f(1+x)$ centered at $x=0$, I need to show that it converges to $f(1+x)$. I got $f(1+x) = \sum (-1)^{n}*x$. I don't know what to do from ...
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1answer
25 views

Abitrary derivatives of lagrange basis functions

The lagrange basis functions are given by \begin{align} \phi_k(x) =\prod_{j\not = k} \frac{x-x_j}{x_k-x_j} \end{align} I try to reproduce the numerical results of a paper. In this paper, the ...
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0answers
35 views

Approximate solutions for quintic equation

The other day I asked a question in here about deriving the equations $$u^2\left(\left(1-s_1\right)+3u+3u^2+u^3\right) =\alpha\left(s_0+2s_0u+\left(1+s_0-s_1\right)u^2+2u^3+u^4\right),$$ where ...
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0answers
60 views

Remainder of a Taylor Series.

I have found the Taylor series to the second order of $g(λ) := f(x + λ(y − x)), $ for g near $ λ = 0 $ and I got: $g(λ) = f(x) + f'(x)(y-x)λ + (1/2!)*(f''(x)$$(y-x)$^2$($λ^2$))$ + ... I am ...
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1answer
20 views

Using Taylor Polynomial for Estimating Error

Need some help in this question! Let $f(x,y)$ be $C^3$ in the open set $A\subset \Bbb{R}^2$ and let $(x_0,y_0)$ be a point of $A$. Proof that there are an open ball $B$ of center $(x_0,y_0)$, with ...
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4answers
283 views

Which expansion of $e$ is more accurate?

We have two forms of $e^x$ $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....$$ and $$e^x=\frac{1}{\displaystyle 1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+....}$$ The second form comes from ...
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2answers
57 views

How to solve limits with Taylor expansion?

I'm in trouble with Taylor series..... how can I solve limits without Bernoulli-de L'Hôpital method?? For example, $$\lim_{x \to +\infty} \frac{x-\sin{x}}{2x+\sin{x}}.$$ The answer, if I'm not wrong ...
3
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1answer
27 views

Taylor's Theorem For Error Approximation

I'm trying to evaluate a function $f(t)$ with a given $t$ value to within 10$^{-5}$. So, if I use Taylor's Theorem : $f(a)+f′(a)1!(x−a)+f′′(a)2!(x−a)2+⋯+f(n)(a)n!(x−a)n.$ Would my $t$ value = $a, ...
4
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1answer
23 views

Equality involving Taylor coefficients

Considering the following series expansion $$ \frac{1}{{1 - 2x - x^2 }} = \sum\limits_{n = 0}^\infty {a_n } x^n $$ prove that $ \forall n,\,\exists m $ such that $ a_n ^2 + a_{n + 1} ^2 = a_m ...
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0answers
18 views

Rational Binomial for Taylor series

I want to write this as a sum while $x_0 = 0$ $$f(x) = e^{-x}(1-x)^{-1/2}$$ I know the sum for $e^{-x}$ but I can't figure out the sum for $(1-x)^{-1/2}$ What I tried (minuses cancel out because of ...
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2answers
30 views

Taylor series example

Task : Write down the McLaurin series till the 4th power ( Hope the translation from german is good :) ). $f(x) = (e^{-x} - 1)^2$ What I did , because I thought that expanding $(e^{-x} - 1)^2$ till ...
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1answer
33 views

understanding phrasing of taylor polynomial question

Show that $|\sin x - x + \frac{1}{6}x^3| < 0.08$ for $|x| \le \frac{\pi}2$. How large do you have to take $k$ so that the $k$th order Taylor polynomial of $ \sin x$ about $a=0 $ approximates $\sin ...
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5answers
65 views

Taylor Series of $2xe^x$

I have to find the Taylor Series for $2xe^x$ centred at $x=1$. I came up with the following. $$e^x = e^{x-1} \times e = e \bigg( \sum_{n=0}^\infty \frac{(x-1)^n}{n!}\bigg)$$ Then consider $2xe^x$. ...
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1answer
22 views

Linear Taylor Polynomials about 0

(a) Find the linear Taylor polynomial about $0$ for $(1 + x)^{15}$ For the first question I tried to use the formula: $1 + px + p\frac{(p-1)}{2!} x^2$ This didn't work for me as when I substituted ...
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0answers
45 views

Multi-variate Taylor Series Expansion

I understand how to use Taylor series to expand basic functions. However, I am trying to work out how to expand Taylor series with more than one variable. So far I have the equation with the two ...
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1answer
53 views

Series representation of $1/|x-x'|$ using legendre polynomials

Given two vectors $\mathbf x\in\mathbb{R}^3$ and $\mathbf x'\in\mathbb{R}^3$. Assume: $x = |\mathbf x|$ and $x' = |\mathbf x'|$. Prove that: $$ \frac{1}{|\mathbf x - \mathbf x'|} = \frac{1}{\sqrt{x^2 ...
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14 views

Systems of equations using taylor's series and find an upper bound

I have a function $g(a)=f_i(x+a(y-x))$ where a$\in$$\Re$ and x,y$\in \Re^d$. I consider $g'(a)=f'_i(x+a(y-x))(y-x)$ and $g''(a)=f''_i(x+a(y-x))(y-x)^2$ Then I have to plug them in the Taylor's series ...
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0answers
48 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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0answers
28 views

Show that $\sin z$ has only one series expansion

The question goes: An extension of the real function $\sin x$ into a complex analytic function is by defining $\sin z = z- z^3/3! + z^5/5!- \cdots$. Show that this is the only way6 to extend $\sin x$ ...
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1answer
30 views

Demonstrate series of Maclaurin

Find the Maclaurin series of $$f(x)=xe^x$$ Integrate this series term by term in the closed interval $[0,1]$ and demonstrate that: $$\sum^\infty_{2} \frac{1}{(n-2){} !n} = 1$$ I tried it: ...
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1answer
91 views

If f ' = 0, then f is constant?

I'm a little confused. After finishing the online multi-variable calculus course from the MIT OCW offerings (I wanted to brush up on the subject more concretely, after my Analysis II course), I ...
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2answers
33 views

Solving a limit with taylor

I'm stuck in solving this limit $$\lim_{x\to0} \frac{(1+x)^{\frac1x} - e}{x}$$. Here I can must use Taylor expansion. My idea is to obtain the form $e^y-1$ on the numerator and then use Taylor ...
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0answers
29 views

Using Taylor's theorem and Lagrange form of the reminder to prove the second order condition for convexity

I try to prove the second order condition for convexity. So far' I've done the following: First, I prove second order => convexity: Let $f$ be a function with positive semi-definite Hessian. Using ...
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0answers
26 views

Expansion of integrand before integration?

I have the following integral as part of a calculation $$\int_{-A}^{A} \int_{-A}^{A} \frac{1}{(z^2 + d^2)^3} dz dx, $$ where $A$ is a constant. I am given the condition $d \gg z$ so I am wondering if ...
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1answer
32 views

How many terms does it take in the expansion of arctan(x) to get pi to 10 decimal places?

I was trying to find a mathematical way to find out how exactly how many terms it takes, but I've no idea. I just know that it's a lot. Thank you!
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1answer
31 views

How to compute the following taylor series expansion

I'm supposed to find the Taylor series expansion of $(\arcsin(x))^2$, but I can't think of a proper solution .The derivative doesn't show much promise since it still contains the $\arcsin(x)$ ...
19
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1answer
124 views

Is it possible to detect periodicity of an analytic function from its Taylor series coefficients at a point?

Given the Taylor series $\sum a_k (x - x_0)^k$ of an analytic function, it is possible to determine whether the function is periodic more-or-less directly from the coefficients $a_0, a_1, \ldots$ of ...
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2answers
235 views

Alternating Series , why start at n = 1?

$$\sum_{n=1}^\infty(-1)^nb_n$$ Convergent if $b_{n+1} \le b_n$ and if $\lim b_n = 0$ I'm learning taylor series now , and I'm confused with this alternating series test , I've searched around and ...
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1answer
45 views

Expansion in large and small limits

Let $$f(x) = \frac{1}{\log(\frac{x}{c})}$$ where $c$ is some constant number. Consider the variable $x$ in the large regime where $x \gg c$ and small regime where $x \ll c$. How would $f(x)$ depend on ...
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0answers
11 views

Taylor Expansion to Approximate Mean of R.V

If X is a random variable with mean $\mu$ and variance $\sigma^2$, how can we use a second-order Taylor expansion (around the mean $\mu$) to approximate the mean (expected value) of a random variable ...