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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How many terms in the series $arctan(x)$ would be needed to get $\pi\ $to the $10$th decimal place?

I got $\pi=\frac 41-\frac 43+\frac 45-\frac 47+\frac 49\ldots$ but I can see that using this it will take me a very long time to reach the decimal expansion I'm looking for. I thought about setting ...
2
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1answer
44 views

Why does the floor function $x \mapsto \lfloor x \rfloor$ have expansion $x + O(1)$?

Shouldn't it just be the largest previous integer? Why is there a remainder term $O(1)$? Thanks, Edit: I am working on a problem that uses the Abel summation formula, and the integration part of ...
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1answer
43 views

Expression for $1 - 2^z x + 3^z x^2 - 4^z x^3 + \cdots$

Using Taylor series we have $$\frac 1 {(1+x)^2} = 1 - 2x + 3x^2 - 4x^3 + \cdots$$ Then multiplying by $x$ and differentiating we get $$\frac {1-x} {(1+x)^3} = 1 - 4 x + 9 x^2 - 16 x^3 + \cdots$$ ...
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0answers
24 views

Taylor series at a certain point converges to the function only at this point.

Find a real valued function on $\Bbb R$ which has derivatives of all orders and whose Taylor series at a certain point converges to the function only at this point. I think $e^{-1/|x|}$ will work ...
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2answers
37 views

Finding the limit of the following series, wherever it is convergent

Let $ f(x) = \sum_{n=1}^{\infty}a_nx^n $ where $a_n$ is the nth Fibonacci number. Find the values for which the series is convergent, and find $f(x)$ for those values. (i.e - find the limit when ...
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1answer
24 views

Taylor series estimation of differential equation

I have a differential equation $$ x'(t) = tx + t^4$$ with initial condition $ x(5)=3$. I am asked to find the estimates using the taylor series method from $o < t < 5$ with $h=0.01$ steps. I get ...
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2answers
80 views

Why does each successive term in a Taylor series need to be much less than the previous term?

This is an extension to this previous question for this original question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works ...
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2answers
70 views

Power series expansion of $x\ln(\sqrt{4+x^2}-x)$

Find $a_n $ where $x \ln(\sqrt{4+x^2}-x) =\sum_{n=0}^{\infty} a_nx^n$. I know that I must find power series expansion of $ln(\sqrt{4+x^2}$ but it doesn't help. Can anyone give me a hint? many ...
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0answers
22 views

quadratic convergence of Newton's method : second derivative

If a function has a zero second derivative at its root, it cannot achieves quadratic convergence? Is zero second derivative equivalent to no second derivative? as from wiki i see when the function ...
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1answer
27 views

Construct Maclaurin series for $f(x)=x\sin(2x)$ in sigma notation and use this to find $f^{(14)} (0)$ and $f^{(9)} (0)$

So I used the known power series of $\sin(x)$ to get down to the Maclaurin in sigma notation. $$\sum_{n=0}^{\infty }\frac{(-1)^{n}(2)^{2n+1}}{(2n+1)!}x^{2n+2}$$ I'm a bit foggy on the $f^{(14)} (0)$ ...
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0answers
17 views

The proof of Newton's method quadratic convergence (Taylor's theorem)

First of all, why do we take the absolute value of both sides? What is the point/reason? and for (x_n - z)^2 , isnt it always positive? Second of all, do you call this a recurrence relation? for the ...
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2answers
32 views

If the first nonzero derivative at $a$ is of odd order $n\ge 3$, then $a$ is a point of inflection

Statement to Prove: Let $f$ be a real valued function such that for a fixed point $a$ , $$f^k(a)=0;1\le k\le n-1;\\and\ \ f^n(a)\neq 0.$$ Then if $n$ is odd then $a$ is a point of inflection. ...
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3answers
180 views

How can I prove that $e^x \cdot e^{-x}=1$ using Taylor series?

When proving $e^x.e^{-x}=1$ by using Taylor series, there are infinite many terms of $e^x$ and $e^{-x}$. Is there any fancy way to combine terms by terms to show that eventually it is equal to $1$?
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2answers
15 views

Hessian at a non-stationary point

I have a function $G(Q) : \mathbb{R}^n \rightarrow \mathbb{R}$ that is known to be convex. I also know that $Q^*$ is a minimum of $G(D)$. If I apply Taylor's theorem to $G(Q)$ at $Q^*$, I get: $$ ...
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0answers
14 views

what's the taylor serie and it's convergence

I have this problem: What is the Taylor series of $\sqrt{x}$ at $x_0 = 4$. What is its interval of convergence? I am stuck and I can not finish it. Any idea on how to do that? Thank you
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0answers
15 views

Questions about the proof of Quadratic convergence with taylor's theorem

First of all, why do we take the absolute value of both sides? What is the point/reason? and for (x_n - z)^2 , isnt it always positive? Second of all, do you call this a recurrence relation? for ...
0
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1answer
47 views

Is my idea of decomposing a meromorphic function into a sum of Laurent series correct?

We know that complex-analytic functions $f(z)$ agree with their power series representations on their domain of analyticity. If a meromorphic function has simple poles at $z_1, ..., z_m$ and ...
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2answers
62 views

Let $\mathbf A$ be a matrix such that $\mathbf A^2=-\mathbf I$. Prove that $\exp(\varphi\mathbf A)=\mathbf I\cos{\varphi}+\mathbf A\sin{\varphi}$

Let $\mathbf A$ be a matrix such that $\mathbf A^2=-\mathbf I$. Prove that $\exp(\varphi\mathbf A)=\mathbf I\cos{\varphi}+\mathbf A\sin{\varphi}$ This is my attempt: $$\mathbf A^2=-\mathbf I ...
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0answers
18 views

Is the proof of the statement make sense?

Please refer this link for some background material http://www.docdroid.net/161p6/curve.pdf.html So i propose a statement to a online tutor, the answer at the below link is the proof of the ...
2
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1answer
35 views

Counter example to theorem in complex domain

A theorem on Taylor series in complex domain is as follow: Suppose $f(z)$ has Taylor series at $a$ with convergence radius of $R$. Then $f(z)$ has at least one singular point on $|z-a|=R$. But I ...
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3answers
36 views

Taylor expansion of a complex function on a disc

I need to find the taylor expansion of the complex function $\frac{z^2}{z-2}$ on the disc $|z|<2$ I'm not sure how to start this off, can anyone help me?
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0answers
37 views

Series expansion of $ (1-\frac{1}{x})^x $ at infinity

I'm trying to compute the expansion of $ (1-\frac{1}{x})^x $ at infinity, which is given by WolframAlpha as $$ \frac{1}{e} - \frac{1}{2ex} - \frac{5}{24ex^2} - \frac{5}{48ex^3} - \frac{337}{5760ex^4} ...
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2answers
46 views

find the interval of convergence of Taylor series

Represent the function $f(x)= x^{0.5}$ as a power series: $\displaystyle \sum_{n=0}^\infty c_n(x−6)^n$ Got that: $c_0$ = $\sqrt{6}$ $C_1=\dfrac{1}{2\sqrt{6}}$ ... But I couldn't find the interval ...
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0answers
22 views

Product with multi-dimensional matrix

We know the power series in scalar case $f:\mathbb R\to\mathbb R$ $$ f(x)=a_0x^0+a_1x^1+a_2x^2+a_3x^3+\dots $$ so what is the extension to the multi-dimensional case $f:\mathbb R^n\to\mathbb R$ $$ ...
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1answer
39 views

Meaning of $C^k$ in Taylor's expansion [closed]

In the following statement, what does $f \in C^k$ mean? And why is there a $q$ for the last part of expansion? So now if I let $k = 2$, what does it mean? And will the expansion involve 3nd ...
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2answers
34 views

How many n derivatives do you take for Taylor series to be accurate? [closed]

How many derivatives must we take to consider some Taylor series an accurate reflection of a function?
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23 views

taylor expansion in 3 variables

I would like to compute the Taylor expansion at the order 2 in (0,0) of the function $g(x,y,z)=x^{17}+xy^2+xyz+3xy+5yz+2x+1$. I know there is a fomula for a 2-variable function which is ...
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1answer
19 views

Coefficients of power series

After expansion, we have $$ (x_1+x_2+\dots+x_n)^m=a_1x_1^m+a_2x_1^{m-1}x_2+\dots $$ where $x_{()}$ is the variable and constant indices $n>m$. What is the expressions of all these possible ...
2
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1answer
44 views

How do you Taylor expand the log likelihood function of the Poisson distribution?

This question is an extension to this previous question asked by myself: When doing a maximum likelihood fit, we often take a ‘Gaussian approximation’. This problem works through the case of a ...
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0answers
32 views

Why is the Taylor series of $1/\sqrt{1-4q^2}$ popping up in my recursively defined triangle of polynomials?

While answering this question I stumbled on some nice (inexplicable) observation where a recursively defined sequence of polynomials turned out to coincide with some Taylor development I'll start ...
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3answers
39 views

Prove the following using Maclaurin's theorem

Prove that $$\log(1+e^x)=\log 2+\frac{1}{2}x+\frac{1}{8}x^2-\frac{1}{192}x^4......$$ I have tried doing it. Tell me if you think the question is wrong
2
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2answers
25 views

Prove the series expansion

Prove that $$(1+x)^\frac{1}{x}=e-\frac{e}{2}x+\frac{11e}{24}x^2-\frac{7e}{16}x^3....$$ where e is exponenial , can any one give a proof...I tried with series expansion i could not get it.
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2answers
234 views

The integral $\int\ln(x-\ln(x))~dx$

The integral $f(y)=\int_0^y\ln(x-\ln(x))~dx$ is on my mind. I'm not sure if this has a closed form? Maybe we need to use the lambert-W function to solve this one? If it cannot be done in closed ...
3
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0answers
46 views

Which version of Taylor Theorem is this?

Suppose $X$ is a random variable and $\psi(t)=E[\exp(itX)]$ is its characteristic function. Let $K(t)$ be the principal value of the logarithm of $\psi(t)$. Suppose further that ...
6
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3answers
87 views

How to show $1 +x + x^2/2! + \dots+ x^{2n}/(2n)!$ is positive for $x\in\Bbb{R}$?

How to show $1 + x + \frac{x^2}{2!} + \dots+ \frac{x^{2n}}{(2n)!}$ is positive for $x\in\Bbb{R}$? I realize that it's a part of the Taylor Series expansion of $e^x$ but can't proceed with this ...
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2answers
45 views

Taylor series of $\ln(x+2)$

I try to determine the Taylor series of $\ln(x+2)$. Since I know $\ln(1-x) = \sum_{n=1}^{\infty} \frac{x^n}{n}$, I suppose I can rewrite, \begin{align} \ln(x+2) &= ...
0
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1answer
24 views

Finding the error in a two-step finite difference numerical approximation

I got the following question in a math lecture the other day, and I'm not really sure how to go about it: A differential equation is given in the form $$\frac{\partial y}{\partial x} = f (x, ...
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1answer
49 views

Taylor series and radius of convergence: $\sqrt{x}$ with centre $x = 16$?

I've been struggling with this question for a while now and getting nowhere with it. Could someone please help me out? Assuming that the function has a power series expansion about the given point, ...
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0answers
28 views

Taylor polynomial approximation with absolute error less than 3 decimal places

Assume that f is a function with $|f^{(n)}(x)| \le 11,$ for all n and all real x. Let $T_n(x)$ denote the Taylor polynomial of degree n for f(x) about the point $x=0$. What is the least integer n ...
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2answers
49 views

Help with arithmetic on basic Taylor Series expansion

There are two of the steps below that I would seek assistance on the arithmetic. From the wikipedia article on Taylor series: https://en.wikipedia.org/wiki/Taylor_series The Maclaurin series for ...
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4answers
68 views

Maclaurin polynomial of tan(x)

The method used to find the Maclaurin polynomial of sin(x), cos(x), and $e^x$ requires finding several derivatives of the function. However, you can only take a couple derivatives of tan(x) before it ...
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2answers
32 views

Maclaurin $f(x)=\sin^4x,x\in R$

Write Maclaurin Polynomial$$T\small{10}(x)$$ for function $$f(x)=\sin^4x,x\in R$$ Maclaurin Polynomial: $$T10(x)=f(0)+f'(0)x+f''(0)\frac{x^2}{2!}+...+f^{10}(0)\frac{x^{10}}{10!}$$ For my problem ...
3
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1answer
59 views

Evaluation or asymptotic for $\int_1^x y\sin\left(\frac{2\pi (y-1) x}{y}\right)dy$

Truly, my genuine problem (see Appendix for context) is compute in a closed form or an asymptotic, for real $x\geq 1$, for $$\int_1^x\left(\int_0^{y-1}\cos\left(\frac{2\pi t ...
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0answers
7 views

Taylor Remainder over an interval for polynomial interpolation

When attempting to find how big n should be so that $|e^x - p(x)| < 10^{-4}$ over the interval $[-1,1]$ using Taylor Remainder, what value should I be using for $x$ in $(x - x0)^{n+1}$? I'm using ...
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1answer
38 views

Why Taylor series “is convergent” to differential when $\Delta x$, $\Delta y$ go to $0$?

Let $f(x,y)$ be a smooth function. Let $\Delta x$ and $\Delta y$ denote small differences in arguments $x$ and $y$, respectively. For any $x_0,y_0$ we can find Taylor series centered at that point: ...
1
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1answer
25 views

Error on Taylor formula argument

Question: My solution: $$f''(x) = \frac{f(x+h) - 2f(x) + f(x-h)}{h^2} $$ $$f''(x) = \frac{1}h \frac{f(x+h) - 2f(x) + f(x-h)}h$$ $$f''(x) = \frac{1}{h} [f'(x)-f'(x) = 0]$$ So because the ...
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1answer
12 views

Second degree multi variable taylor polynomial

Let f (x, y ) = x cos(πy ) − y sin(πx) point: 1,2 I am following the standard formula, which starts with taking the partial of f with regards to x twice, which gives me: ysin(πx)π But plugging in ...
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1answer
31 views

Upper Error Bound Taylor Series

(a) Given $f(x) = \sqrt{x}$, find its Taylor polynomial of degree 2 centered at $x=4$ and use it to estimate $\sqrt{5}$. (b) Use Taylor's theorem to give an upper error bound for the estimate in part ...
0
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4answers
58 views

Proof of “Taylor Series”

We know that , according to the Taylor Series : $$f(x)=f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+ \cdots \tag{1}$$ And then Maclaurin series as $a=0$. ...
1
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1answer
24 views

Evaluating integral using invalid substitution

I was trying to show that for suitable t: $$ 2\pi(1+t/(\sqrt{(1-t)(3-t)})=\sum_{0}^{\infty}(t^n\int_0^{2\pi}1/(2-cos(\theta))^nd\theta $$ By uniqueness this is clearly the Taylor series about $0$ ...