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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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
votes
1answer
22 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
15 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
26 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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0answers
21 views

unique combinations of A,B,C? [on hold]

If you have 3 solutions of A,B,C, - how many unique combinations can you make using only 2 at once? If I had, say, a 3 solutions labelled A, B and C - how many unique solutions can I make using only ...
1
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1answer
23 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 ...
3
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5answers
56 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 $2xe^x$. ...
0
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1answer
21 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
39 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 ...
0
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2answers
39 views

Using Taylor series to find $\lim_{x\to 1}\frac{2-(x+3)^{1/2}}{x-1}$ [on hold]

How to find the limit $$\lim_{x\to 1}\frac{2-(x+3)^{1/2}}{x-1}$$ using the Taylor series? I have done the derivatives on function following the regular steps. But all I get is zero.
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0answers
12 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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0answers
9 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 ...
3
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0answers
28 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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vote
0answers
26 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 $\sin x$ ...
0
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1answer
28 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: ...
3
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1answer
76 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 ...
0
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2answers
30 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 ...
0
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0answers
16 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
24 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 ...
2
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1answer
28 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
28 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 $\arcsin(x)$ ...
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0answers
22 views

Taylor Method ambigous

As far as I know that Taylor method goes like this Now I've got this problem Why did we solve it like $\cos(x-\frac{\pi}{2}+\frac{\pi}{2})$, why not $\cos(\frac{\pi}{2})$? DISCLAIMER: I am not ...
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0answers
14 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 ...
5
votes
2answers
222 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 ...
0
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1answer
27 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 ...
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votes
2answers
21 views

maclaurin series for function undefined at a point

Say i want a power series for a function such as $$\frac{(2x+2)(x)}{(2x)(3x+1)}$$ at $x=0$. How would one go about this? I have acquired the second, third and fourth terms, but am struggling getting ...
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0answers
11 views

Forward-difference approximation using taylor expansion

Consider a Forward-difference approximation for the second derivative of the form, $f''(x) = Af(x) + Bf(x+h) + C(x+2h) $ Use Taylors Theorem to determine coefficients, $ A,B, C$ that give the ...
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votes
3answers
39 views

Expanding $(1 - x + 2y)^3$ in powers of $x-1$ and $y-2$ with a Taylor series

I would like to do this. I observe that I can write $$f(x,y) = (1 - x + 2y)^3 = (2(y-2) - (x-1) + 4)^3.$$ It's easy to do this via algebra directly. However, I'm asked to do it by computing the ...
0
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1answer
38 views

If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$

Prove: if $f''(x_0)$ exists then $\lim\limits_{x \rightarrow x_0} \dfrac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$. I'm not exactly sure how Taylor's theorem fits into all this, but I found ...
1
vote
1answer
27 views

Finding Taylor series of $x^{-3}$ about $x=a$

How to find the Taylor series of $x^{-3}$ about $x = a$? Usually I can do ones where $f(x) = (x+c)^{-3}$ but when $c=0$, I'm unsure. Even for positive exponents there's a simple way.
4
votes
1answer
22 views

Remainder in the multivariate Taylor expansion

For the function $f:\mathbb{R}\to\mathbb{R}$, I can write the Taylor expansion $$f(x+h) = f(x) + f'(x)h + \frac{1}{2!}f''(x)h^2 + O(h^3)$$ where the remainder is $o(h^2)$ as well. I'm confused with ...
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0answers
51 views

Taylor's Theorem for Multivariable Implict Functions

I'm trying to find the $2$nd order Taylor polynomial for $z=g(x,y)$ near the point $(\frac {\pi}{2}, 1,1)$, given the function $\sin(xyz)=z^2$. I've never found the Taylor polynomial of a function ...
1
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1answer
17 views

Bound remainder of Taylor series with Lipschitz property of derivative

I feel like this should be fairly simple: I would like to use the fact that $$|g'(x) - g'(y)|\leq C|x-y|^\delta$$ for all $x,y\in \mathbb{R}$ for some $C,\delta >0$ to put a bound on the ...
0
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1answer
27 views

Using Taylor series to create a zero function

Let $f(t)$ be a $n$th order polynomial with real, positive coefficients (I am not sure if these constraints are necessary). Then after taking $n+1$ or more derivatives, the function vanishes and is ...
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0answers
23 views

Order of zeros for the function f(z) = e^z -1

I need to find the order of each of the zeros for the function $f(z) = e^z -1$. I've used the trig identity for $e^z$ to determine that the zeros are at $z_0 = 2n\pi i$ where $n \in \mathbb{Z}$. I ...
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votes
1answer
35 views

Find the Maclaurin series [closed]

Could anyone help me find the Maclaurin series expansion for $f(x)=e^{2x-x^2}$ up to $ x^3$, using table expansion for $e^x$? Thank you in advance.
3
votes
1answer
26 views

Equivalence of two expressions

I have the following expression $$\varphi = -5\Delta t + \sqrt{25(\Delta t)^{2}+1}$$ and I want to show that in fact this is equal to $$\varphi = e^{-5\Delta t} + \mathcal{O}(\Delta t^{3}).$$ To do so ...
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2answers
46 views

Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.

I was google-ing and came across this question. Till now I don't have any solution. Let $f$ be a double differentiable function on $(1,\infty)$. Let $M_0=\sup_{x\in R}|f(x)|$, $M_1=\sup_{x\in ...
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1answer
32 views

uniformly continuous when second derivative is bounded

Let $f$ be continuous on $[a,b]$. $f$ is twice differentiable on $(a,b)$ and $|f^{\prime\prime}(x)|\leq M$ for all $x\in [a,b]$. Show $f$ is uniformly continuous. This is a question from an exam in ...
3
votes
3answers
26 views

Finding the $n$th Taylor coefficient of $g(z)=\frac{z}{(z-b)^2}$ centered at $a$ (where $a=2-\sqrt{3}$ and $b=2+\sqrt{3}$?

I've introduced $a$ and $b$ in order to simplify the notation : $a=2-\sqrt{3}$ and $b=2+\sqrt{3}$. I'm trying to compute the Taylor Series for $g(z)=\frac{z}{(z-b)^2}$ centered at $a$. I denote the ...
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0answers
24 views

Let $Tn(x)$ be the Taylor polynomial of degree up to n for $y = sin x$ expanded about $0$. Find an $n$ such that $|sin(0.1) − Tn(0.1)| ≤ 10^{-6}$

Show that your value $n$ works. This is an in-class calculus exercise. My solution: Let $Rn(x) = |sin(0.1) - Tn(0.1)$ $|Rn(x)| <= 10^{-6}$ $|Rn(x)| <= k/(n+1) * |x|^{n+1}$ where ...
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0answers
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$f:[0,1] \to \mathbb{R}$ with $f(0) = 0 = f(1)$ and $|f''|\leq M$. Show $|f'(1/2)| \leq \frac{M}{4}$. [duplicate]

I attempted a solution using this version of Taylor's Theorem, \begin{align*} |f(d) - P_n(d)| &= \left| \frac{f^{n+1}(t)}{(n+1)!} \right| |d-c|^{n+1} \\ |f(d) - f(c) - f'(c)(d-c)| &= \left| ...
3
votes
1answer
41 views

Find a polynomial which approximates $f(x) = \sqrt{x}$ in the interval $(4,5)$ within $10^{-8}$

Find a polynomial which approximates $f(x) = \sqrt{x}$ in the interval $(4,5)$ within $10^{-8}$. I've been trying to solve this using Taylor's remainder theorem, $$ |f(d) - P_n(d)| = \left| ...
1
vote
1answer
97 views

calculating the taylor term of an integral

an exercise ask me to calculate the Taylor term at $x = 0$ and degree four. I know how to take a derivative of an integral, but I'm having doubts about this one. The function: $$\int_0^x e^{-t^2} ...
1
vote
1answer
21 views

Taylor series of an implicit function

Suppose the function $s:[-\delta, \delta] \to \mathbb{R}$, $\delta > 0$, is defined implicitly by $$s(t) = 1 - c\beta t (s(t))^{\beta}$$ for some $c > 0$, $0 <\beta < 1$. Can an ...
4
votes
0answers
33 views

Manipulation of Taylor/Laurent series

I have a question regarding how to expand a given rational function into its Taylor/Laurent series representation. Suppose we are given the function $$f(z) = \frac{z}{(z-1)(z-3)},$$ and are asked to ...
2
votes
1answer
38 views

Evaluate Infinite Sums

I'm having trouble with this question. $$E=\sum_0^\infty \frac{nhve^{-nx}}{\sum_0^\infty e^{-nx}}$$ where x=(hv)/(kT) Evaluate both sums and show that E=hv(e^x -1)^-1 I've tried comparing ...
1
vote
1answer
25 views

How do I perform taylor expansion of the following?

Taylor expansion about $(x,y)$ of $f(x + a,\; y + k\; f(x + b,\; y + c))$ I do not understand what happens to the second $f$ inside. The inspiration for this question is Runge-Kutta methods.
3
votes
2answers
31 views

Showing that the integral remainder of the Taylor expansion of $f(x)=-\log(1-x)$ goes to $0$

Let $|x|<1$. Define $R_n(x):=\int_{0}^{x}\frac{(x-t)^{n-1}}{(1-t)^n}dt$. How do we prove that $\lim_{n\to \infty}R_n(x)=0$? This is actually the integral remainder of the Taylor expansion of the ...