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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2
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53 views

Could someone check my solution for finding constant of a difference quotient?

So the question was, Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be three times differentiable and $f'''$ is bounded, find constants $a,b,c$ such that $$f''(x) = \lim_{h\rightarrow 0} ...
-2
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0answers
35 views
2
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1answer
37 views

Computing the Taylor expansion of the square root of cos(z),

Let $\large f(z)=\sqrt{cosz}$ with the branch of the square root chosen so that $f(0)=1$. Consider the power series expansion of $f(z)$ in powers of $z$. Part 1) Compute the first three non-zero ...
0
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0answers
13 views

Bounding an expression

I am trying to figure out an upper bound on the following expression $$(1 + \epsilon)^{\frac{A}{1+\epsilon} - B}$$ where $\epsilon \in (0,1)$, $A \in (0,1)$ and $B \in \{0, 1\}$. I tried doing the ...
0
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1answer
24 views

How to show that $1+ \sum \limits _{n=1} ^\infty \frac {x^n} n$ converges pointwise?

I am having trouble showing that the taylor series for $-\ln(1-x)$ converges pointwise on $[0,1)$. I have that the $k$ derivative is $\dfrac {(k-1)!} {(1-x)^k}$. This gives that the Taylor series ...
1
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0answers
21 views

Taylor series Integral

When can we use Taylor series expansion and write $\int_0^{\infty} \log(f(x+\alpha x)) dx = \int_0^{\infty}\log(f(x)+\sum_{n=1}^{\infty}\frac{f^{n}(x) (\alpha x)^n}{n!}) dx$? I think, first the ...
2
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0answers
26 views

Suppose that $|f(z)| \leq e^{-1/|z|}$ for all $z\neq 0$. Prove that $f=0$.

Suppose f is entire function such that $$|f(z)| \leq e^{-1/|z|}$$ for all $z\neq 0$. Show that $f=0$. Hint: Consider the Taylor series of $f$ about $0$ and recursively show that all coefficients ...
2
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2answers
91 views

Definite integral of $e^{x/2}$ using Maclaurin polynomial

My professor asked us to find the 3rd degree Maclaurin polynomial of $e^{x/2}$ which I found to be $$1 + \frac{x}{2} + \frac{x^2}{8} + \frac{x^3}{48}$$ I do know that that the series for ...
0
votes
2answers
54 views

Find Taylor Series of $\frac{1}{1+z^2}$ around $1$

For $f(z)=\dfrac{1}{1+z^2}$ find the Taylor series centered at $1$. While I know I could use partial fractions or perhaps maneuver this problem by adding constants, I would really like to use the ...
0
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1answer
15 views

Analysis: Calculate the Taylor Series and determine radius and interval of convergence

This is the function: $f(x)=e^{3x}$ and I am required to calculate it's Taylor series about $a=-2$. I am also required to determine the radius and interval of convergence of the resulting power ...
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0answers
16 views

When $0\le h \le 0.01$, show that $e^h$ may be replaced by $1+h$ with an error of magnitude no greater than $0.6$% of h.

When $0\le h \le 0.01$, show that $e^h$ may be replaced by $1+h$ with an error of magnitude no greater than $0.6$% of h. use $e^{0.001} = 1.01$ What I did was :-
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votes
2answers
52 views

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 ...
1
vote
2answers
37 views

Complex Equation Formula

Can someone show me how the following two expressions are equivalent: $$\Gamma = \frac{i X - R_c}{i X + R_c} = -e^{-i 2 \mathrm{tan}^{-1} (\frac{X}{R_c})}$$ I'm working through a calculation and I ...
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 ...
1
vote
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 $$\frac {1-x} {(1+x)^3} = 1 - 4 x + 9 x^2 - 16 x^3 + \cdots$$ ...
0
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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 ...
1
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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 ...
0
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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 ...
0
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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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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 ...
0
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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 ...
2
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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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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
votes
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: $$ ...
2
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2answers
78 views

Branch cut for $\sqrt{1-z^{2}}$ and Taylor's expansion!

I'm working in a problem that involves the equation $$ w(z)=\sqrt{1-z^{2}} \,\, . $$ I already know that there're two branch points in this equation, namely $\pm 1$, so there's a Riemann surface ...
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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 ...
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 ...
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?
1
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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 ...
0
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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
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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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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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?
2
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1answer
52 views

Solution of the functional equation $g(x)g(z) = g(x+z)+g(x-z)$

What is the solution for the following functional equation? $g(x)g(z) = g(x+z)+g(x-z)$ The solution given is: $g(z) = 2\cos(z)$. In the derivation of the result (using Taylor expansion), there is ...
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0answers
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
23 views

Meromorphic Function on Extended Plane

How do I prove that every meromorphic function on the extended plane is a rational function?
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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 ...
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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 ...
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 ...
2
votes
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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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
6
votes
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 ...
0
votes
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
votes
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, ...
3
votes
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 ...
0
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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 ...
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, ...