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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Maclurin series for $\sin^2(x)$

I am trying to find the maclurin series expansion for $\sin^{2}x$. First I used the half angle identity: $$\frac{1-\cos(2x)}{2}=\sin^{2}x$$ Then substituted in the maclurin series for $\cos(2x)$ to ...
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1answer
47 views

Maclaurin's Series for $\sec(x)$ with help of Maclaurin's series for $\tan(x)$

Is there any way to derive Maclaurin's series for $\sec(x)$ with the help of Maclaurin's series for $\tan(x)$? As we know, the Maclaurin's Series for $\tan(x)$ is: ...
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1answer
28 views

Taylor Series Exercise

What method should I use to find the Taylor series of $f(x)=\frac{x+2}{2-3x}$ with center 2? Here's what I did: Let $y=x-2$ ...
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1answer
51 views

Real analytic way to explain why the radius of convergence of $1/(1+x^2)$ is small

For any series expansion of $\frac{1}{1+x^2}$, the disc of convergence is blocked by the two singularities on $+i$ and $-i$. A series expansion about $0$ gives a radius of convergence of $1$. Is ...
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1answer
67 views

Finding a function that is harmonic in an annulus,

The problem statement is: Suppose that the real series $∑_0^{∞} a_n$ and $∑_0^{∞} b_n$ converge absolutely. Part 1 Prove that there is a function $u(r,θ)$ which is harmonic in $1<r<2$ and ...
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1answer
29 views

Taylor expanding $f(y+\epsilon U1 + {\epsilon}^{2} U2,t,\epsilon)$ in $\epsilon$

How would one Taylor expand $\epsilon f(y+\epsilon U1 + {\epsilon}^{2} U2,t,\epsilon)$ in $\epsilon$? Somehow the professor obtained the first few terms to be: $\epsilon f(y+\epsilon U1 + ...
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2answers
36 views

How do I calculate the error bound for a Maclaurin series?

How many terms of the Maclaurin series of $f (x) = \ln(1 + x)$ are needed to compute $\ln(1.2)$ with an error of at most $0.0001$?
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1answer
41 views

How to find the Maclaurin series for $f(x) = \frac{1}{1 + \sin(x)}$?

I have that $\frac{1}{1 + x} = 1 - x + x^2 - x^3 + ...$ So then $\frac{1}{1 + \sin(x)}$ should be $ 1 - \sin(x) + \sin^2(x) - \sin^3(x) + ...$ but clearly this is not the case. So how does ...
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1answer
37 views

complex series expansion for $f(z)=\frac{1}{z-1}$

Expand the function $f(z)=\frac{1}{z-1}$ as as a series around $z_{0}$ in two regions a) $$|z-z_{0}| < |1-z_{0}|$$ b) $$|z-z_{0}| > |1-z_{0}|$$ and find coefficient $a_{n}$ is each case. I ...
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2answers
47 views

Power Series Approximation of ln(x)

I am working on building a small embedded calculator, and am working on adding a natural logarithm function that utilizes only + and -. I have worked out the power series representation of ln(x) as ...
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1answer
26 views

Why can the first term of the Taylor series expansion of $\cos(\theta)$ be written in the way below?

Why can the first term of the Taylor series expansion of $\cos(\theta)$ be written as $\cos(\theta_0) - (\theta - \theta_0) \sin (\theta_0)$?
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3answers
87 views

Why does the expansion of $e^x$ appear to arise in the formula for derangement of $n$ things $D_{n}=n!\sum_{k=0}^n \frac{(-1)^k}{k!}$

I was recently toying with wolframaplha with the expansion of $e^x$ and I noticed a strange thing that on keeping $x=-1$ (if it is allowed!!!).. I get on the RHS a strange looking infinite expression ...
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2answers
26 views

Taylor approximation and composition

I have a general and a specific question about the composition of Taylor series. Let's say we have $f(x)$ and $g(x)$. We know that the normal composition of functions is something like this: $g ...
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2answers
54 views

If $f \in C^{\infty}$ and $f^{(k)}(0)=0$ for all integers $k \ge 0$, then $f \equiv 0$.

I thought this was true since, $f(x)=f(0)+f'(0)x+f''(0) \frac {x^2}{2!} + \dots$ But I am wrong. Where did I make mistake?
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0answers
36 views

Show that $\exists \xi\in(0,1)$ satisfying certain condition

Suppose $f\in C^3[-1,1]$ with $f(-1)=0,f(1)=1$ and $f'(0)=0$. Show that for any $a\in\mathbb{R}$, there eixsts $\xi=\xi(a)\in(0,1)$ only depending on $a$ such that $$f'(\xi)-1=a(f(\xi)-\xi).$$ It ...
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1answer
88 views

Conditions of the Taylor Theorem

I'm confused on the assumptions behind the Taylor Theorem because I found different versions of them across several books. Consider the function $f:\mathbb{R}\rightarrow \mathbb{R}$ (1) If and only ...
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1answer
38 views

Strengthened version of Taylor's theorem?

Let $f$ be a continuous real-valued function on $[a,b]$ that is $n+1$ times differentiable on $(a,b)$ and such that $f^{(1)}, f^{(2)},\ldots,f^{(n+1)}$ are bounded on $(a,b)$ and ...
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3answers
56 views

Approximation by using Taylor Polynomials - why?

Could anyone tell me why would I want to approximate a function $f$ by using its Taylor expansion (is it the same as saying approximation by Taylor polynomials?), if I have the exact formula of the ...
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2answers
54 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} ...
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1answer
38 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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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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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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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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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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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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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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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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 ...
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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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2answers
79 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 ...
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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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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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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 ...