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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2answers
102 views

What does the constant mean in Big O notation?

I have a big issue in understanding the real meaning of Big O notation. Classical definition: $f(x) = O(g(x))$ as $x\rightarrow k$ if there exist $\delta, C > 0$ such that $f(x) \leq Cg(x)$ ...
0
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1answer
93 views

Arctan(f(x)) is almost the same as Erf(f(x)) for many f(x). Is the just coincidence or is there a reason?

For example: Arctan(x) is almost Erf(x) (subtle differences in absolute value and curve) Arctan(x^50) is almost Erf(x^50) (difference in absolute value) and many others, so we can conclude: ...
1
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1answer
24 views

Find the series expansion of $\left(e^{(x-1)}\right)^2$

Find the series expansion of $\left(e^{(x-1)}\right)^2$. I thought maybe I could use binomial expansion but that is only for $(1+x)^n$, so now I am unsure how to proceed. I could set $(x-1)^2=n$ and ...
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2answers
25 views

Is there a mistake in this page on asymptotic expansions?

I think there is an error in section 4.3 of this page - http://aofa.cs.princeton.edu/40asymptotic/ The author says that by taking $x = -\frac{1}{N}$ in the geometric series $\frac{1}{1-x} = 1 + x + ...
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0answers
39 views

How do I solve for the elements of the partial derivative of a Hessian matrix?

In the paper about Speeded-Up Robust Features, it says that in order to localize points, interpolation of nearby data is needed to find the location in space and scale. This is done by fitting a 3D ...
1
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0answers
30 views

Will this method find the taylor expansion of ANY function $f(x)$?

Polynomials are themselves Taylor expansions, correct? ex. $4x+5x^2+3 = 3+4x+5x^2 +0x^3 +0x^4 + \dots$ I'm assuming has no closed form besides $\sum_{n=0}^{2}(3+n)x^n + \sum_{n=3}^{\infty}0x^n$ but ...
2
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0answers
25 views

Upper bound on derivatives of very high order?

I am doing a calculation where I am estimating a value $\omega$ by a Taylor polynomial. I know that $\omega \cdot a = f(b)$ and thus I can estimate $\omega$ by $a \cdot T_n f(b) $ where $T_n$ is the ...
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5answers
242 views

Why are Maclaurin series useful if we can only use them for such a small range of numbers?

Okay, I am beginning to get how Maclaurin series work, but what I don't understand is why they are useful. Why would you want an infinite expansion for a series that works for such few values (only ...
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0answers
27 views

The error function in the taylor's theorem for taylor series

I was reading taylor's theorem at wikipedia and at some point they say that $f(x)$ can be written as a function related to its linear approximation $P_1 = f(a) + f'(a)(x-a)$. This is a very simple ...
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0answers
25 views

If I have an infinite series, how do I know that the digits I calculated are rigth?

For example, there are infinite series for $\pi$, $e$, $\phi$... But if I sum a finite ammount of terms, I get an approximation for the series. How do I know how much correct digits of this ...
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1answer
17 views

Singularity and residue in z = 0

How can I classify the singularity in $z = 0$ and determine the respective residue in $z = 0$ for the following function ? f(z) = $ cos(1/z)(z+1)^2$ Do I have to use Taylor expansion of $cos(1/z)$ ...
0
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2answers
36 views

Cannot expand $\sin(2x^2-4x+3)$ at $x_0 = 1$

Trying to expand $\sin(2x^2 - 4x+3)$ at $x_0 = 1$ to the $O(x-x_0)^n$. After substitution $t = x - 1 $, the problem becames $$\sin(2t^2+1) \text{ at } t_0 = 0$$ While we know that $$\sin(s) = ...
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3answers
512 views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
1
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4answers
254 views

Maclaurin series expansion of $\frac{1}{(1+x)^n}$

I am trying to figure out the Maclaurin Series expansion of the function, preferribly in a sneaky and clever way. Any ideas? Thanks.
1
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1answer
66 views

Taylor Polynomial of $f(x)=\cos(x)\cdot\sin(x)$

How would I calculate the third maclaurin/taylor polynomial on $\cos(a) \cdot \sin(b)$, Do I use the product rule when I calculate the derivatives? I don't know where to start or read about it, been ...
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5answers
102 views

When $a\to \infty$, $\sqrt{a^2+4}$ behaves as $a+\frac{2}{a}$?

What does it mean that $\,\,f(a)=\sqrt{a^2+4}\,\,$ behaves as $\,a+\dfrac{2}{a},\,$ as $a\to \infty$? How can this be justified? Thanks.
1
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2answers
29 views

Importance of the first term in a Taylor series

Suppose you have a function $f(x)$ whose Taylor series can be represented as the power series $$a_0 + a_1x^2+a_2x^4+...$$ If you are told that for $x\in\mathbb{R}_+$, $$a_0 + a_1x^2 + a_2x^4 + ...
1
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1answer
36 views

Finding a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$.

I'm trying to find a Taylor Series representation of $f(x)=\ln(\frac{1+2x}{1-2x})$ centered at $0$. So I am using the Maclaurin Series representation of $f(x)=\ln(1+x)$ which is ...
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1answer
40 views

Multiple representations of ternary expansions of numbers

$x \in [0,1]$. If in binary expansions ie series $\displaystyle x = \sum_{i=1}^{\infty} \frac{x_i}{2^i}$ where each $x_i \in \{0,1\}$ we identify the sequences $\underline{x}$ and $\underline{x}'$ ...
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1answer
21 views

Use the power series representations of functions to find the taylor series of $\frac{1}{5+x'}$ at center = -6.

I am trying to find the taylor series of $f(x)=$ $\frac{1}{5+x'}$. And I cannot seem to get how to find the taylor series using the method I've been using for other functions. Another thing that's ...
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1answer
39 views

MacLaurin of the Third-degree in sin(a*x)*cos(b*x) at given values

Alright so from my understanding MacLaurin is a special case of Taylor Series but at f(0). However my question involves solving the third degree of MacLaurin for $$f(x) = sin(a \times x)\times ...
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2answers
33 views

Sum Representation of log(1 + x)

$\log(1+x) = \sum_{k=1}^{\infty} \left(\dfrac{x}{1+x}\right)^{k} \dfrac{1}{k} = \sum_{k=1}^{\infty} \left(1 - \dfrac{1}{1+x}\right)^k \dfrac{1}{k}$ Why is this true? The most sum representation of ...
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1answer
21 views

Expand with a Taylor formula $\frac{2+x}{x^2+2x+2}$ near the $x_0 = -1$

I am not sure whether I am doing it correctly. So, $$\frac{2+x}{x^2+2x+2} = \frac{2}{x^2+2x+2} + \frac{x}{x^2+2x+2} = F_1 + F_2,$$ $$x^2+2x+2 = (x - x_1)(x-x_2), \text{where} \\ x_1 = i+1, \\x_2 = i ...
0
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0answers
38 views

The remainder of a Taylor Polynomial.

I am looking at a problem with Taylor series, and I'd just like to know if I am doing it correctly, or at least headed in the right direction. I start by finding the Taylor series for $arcsin(x)$ ...
1
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2answers
27 views

Simplification of a series so that it converges to a given function

I am trying to rearrange the series $ \frac{1}{1-z} - \frac{(1-a)z}{(1-z)^2} + \frac{(1-a)^2z^2}{(1-z)^3} - \cdots$ In such a way that I can show it converges to $\frac{1}{1-az} $ What I ...
1
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1answer
39 views

The $n$th-derivative of $q(x) = x^4 - 8x^3 - 4x^2 + 3x - 2$, where $n \le 4$

Some factors will be $\frac{4!}{(4-n)!}\cdot a_nx^n, \frac{3}{(3-n)!} \cdot a_{n-1}x^{n-1}, \ldots, \frac{1!}{(1-n)!} \cdot a_0x^0$, but the lowest degree one will always become zero in the next ...
0
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0answers
37 views

Why is it true that $\forall b\in(0,1): (1-b)\left(e(1-b)\right)^{\frac{b}{1-b}}\geq\prod\limits_{n=2}^{\infty}n^{-b^n}\geq 0$

Why is it true that $$\forall b\in(0,1)$$ $$1\geq(1-b)\left(e(1-b)\right)^{\frac{b}{1-b}}\geq\prod\limits_{n=2}^{\infty}n^{-b^n}\geq 0$$ Note: Let $$f(x)=\prod\limits_{n=2}^{\infty}n^{-b^n}$$ Then ...
0
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1answer
55 views

Help finding n-order Maclaurin polynomial

EDIT AND PLEASE NOTE: I DON'T want solutions that are nicer or more elegant but presume knowledge of other infinite series and/or don't come from the nth-derivative because I'm precisely studying how ...
1
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1answer
20 views

Can anyone explain how to show the finite difference equation $y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$?

I was given that $y_{j}=y(x_{j})$ where $x_{j}=x_{0}+jh$ for integer j and positive h. I need to show that $$y'_{0}=\frac{y_{1}-y_{-1}}{2h} + O(h^{2}).$$ I thought I could start by finding the Taylor ...
2
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0answers
64 views

Book Request: Taylor's Theorem for functions $f: \Bbb R^n \to \Bbb R^m$

I'm looking for a resource (e.g. a book, website, or arxiv paper) that goes over the general case of Taylor's theorem, with a full proof and examples. Do you guys know of any material that covers ...
3
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1answer
32 views

Are the Taylor polynomials of a function the results of a minimization problem?

Here is an example to better explain my question. Consider the function $f(x) = \cos(x)$. I want to approximate it in the set $[-\pi; \pi]$ using a polynomial $g(x) = a + bx + cx^2$ of order $2$. ...
1
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1answer
79 views

Laplace equation, Taylor expansion

I couldn't find it anywhere, so I decided to write my question here: I have problems solving this equation: $$u_{xx} + u_{yy} = 4,$$ subjected to the conditions $$u(x,x)=2x^2, \quad u_x(x,x)=2x$$ ...
2
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3answers
59 views

Why is this true? $\forall a\in(1,\infty), B\in(0,\infty), x\in(0,\infty) : a^x\geq \left(\frac{ex\ln(a)}{B}\right)^{B}$

I know $$\forall a\in(1,\infty), B\in(0,\infty), x\in(0,\infty)$$ $$a^x\geq \left(\frac{ex\ln(a)}{B}\right)^{B}$$ can be proved using AM-GM. Is there a simple way to show the inequality holds in all ...
2
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2answers
40 views

Approximation of a ratio

Is this approximation true? If so, why? $$\frac{1+x}{1+y}\approx 1+x -y$$ I think it has something to do with $x$ and $y$ being close to zero, so that the ratio of the two is approximately equal to ...
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1answer
46 views

The accuracy of approximating $ f(x) = x^{2/5}$ for $0.9 \le x \le 1.1$ using the cubic Taylor polynomial

For the equation $ f(x) = x^{2/5}$, $a=1$, $n=3$, $0.9 \le x \le 1.1$ I was able to approximate f by the following Taylor polynomial: $$ F_3(x) = 1 + \frac2 5 (x-1) - \frac3{25}(x-1)^2 + ...
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0answers
13 views

Symmetry of convergence around a of a Taylor Polynomial

I heard the lecturer mention this shortly, so I'd like a more detailed explanation of it. He showed us a taylor approx. of ln(x) at x = 1, and showed that it only matches in the interval ]0,2], so ...
2
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2answers
29 views

Taylor series of polynomial.

I know that the taylor approx. of a polynomial centered at 0, if n gets big enough, is just the polynomial itself. But why do people always say "centered at 0"... wouldn't we also get the polynomial ...
6
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0answers
131 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider ...
2
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2answers
72 views

Calculators using Taylor polynomials?

I've always heard that calculators (TI-84's and the like) use Taylor polynomials to approximate trigonometric/exponential/etc functions. Do any of you know this for a fact?
2
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1answer
136 views

The Laurent series of $ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1}$

How to find Laurent series of g(z) ? $$ g(z)=\frac{z^n+z^{-n}}{z^2-(a+\frac{1}{a})z+1} \hspace{10mm} \begin{cases} n \in N \\ 0<a<1 \end{cases} $$ answer is : $$ ...
12
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7answers
4k views

How are the Taylor Series derived?

I know the Taylor Series are infinite sums that represent some functions like $\sin(x)$. But it has always made me wonder how they were derived? How is something like ...
0
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1answer
48 views

Maclaurin series accuracy

Find an $n_1$ such that the $n_1$th-order Taylor polynomial for $\sin(x)$ about $x=0$ gives an approximation of $\sin(x)$ with an error of less than $5\cdot 10^{-10}$, for all $x$ between $0$ and ...
0
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1answer
13 views

Bounding $|n^2(1-(\cos\frac{1}{n})^2)|$

I'm working on a a problem that involves me needing to give an upper bound for the following expression: $|n^2(1-(\cos\frac{1}{n})^2)|$ My attempts at bounding it: Expanding the expression: ...
32
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13answers
31k views

What are the practical applications of the Taylor Series?

I started learning about the Taylor Series in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series. Question: ...
0
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1answer
59 views

On the multivariable Taylor expansion

Apparently the second order multivariable Taylor expansion is: $$f(\mathbf x+\mathbf h)=f(\mathbf x)+ \partial_i f(\mathbf x) h_i + \frac 12 \partial_j \partial_i f(\mathbf x + t \mathbf h) h_i h_j$$ ...
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2answers
77 views

Taylor series of Infinitely differentiable function with nonnegative derivatives

Let $f(x)$ be a nonnegative and infinitely differentiable function on $[-a,a]$ to $\mathbb{R}$ such that $\forall x\in[-a,a]:f^{(n)}(x)\ge0$. Prove that the series: $$\sum_{i=1}^\infty ...
0
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2answers
45 views

Proof of an inequality using Newton's Method

Question: Show that the function $f(x):= x^3 -2x -5$ has a zero $r$ in the interval $I:= [2,2.2]$. If $x_1 :=2$ and if we define the sequence $(x_n)$ using Newton's procedure, show that $|x_{n+1} -r| ...
1
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6answers
40 views

Taylor expansion square

Consider the following expansion $$\sqrt{1+x} = 1 + \dfrac{1}{2}x - \dfrac{1}{8}x^2 + \dfrac{1}{16}x^3 .. $$ Show this equation holds by squaring both sides and comparing terms up to $x^3$. I ...
1
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1answer
61 views

Is there a difference between $a \cdot a^T$ and $a^2$?

The title says it all... I can't see the difference between $a \cdot a^T$ and $a^2$, when $a$ is a vector. However I encountered a formula stating $$\frac{1}{|y+a|} = \frac{1}{|y|} - \frac{y \cdot a ...
1
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0answers
44 views

Functions f(x) equal to Taylor series vs Fourier series vs Bessel series

(I had trouble phrasing the question below due partially to the fact that Bessel functions $J_{\alpha}(x)$ and $U_{\alpha}(x)$ are defined for any complex $\alpha$, so below I tried to express an ...