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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Using Maclaurin series, finding the value of an infinite sum…

I want to find $$\sum\limits_{n = 1}^\infty {{{{{({1 \over 2})}^n}} \over {n(n + 1)}}} $$ The book that has this problem in says to use the Maclaurin series for $(1 - x)\log (1 - x)$. I don't ...
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94 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: $$\tan(x)=x+\frac{x^3}{3}+\frac{...
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1answer
22 views

Expansion of $x \log \left(\frac{l+x}{x} \right)$ about x=0

I've read that $$x \log \left(\frac{l+x}{x} \right)=x \log \frac{l}{x} + O(x^2).$$ I tried to derive this using the usual Taylor series method but kept getting a division by zero. Could anyone ...
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1answer
27 views

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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36 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 + {\epsilon}^{...
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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 $y=x-2$ $f(x)=-\frac{y+5}{3y+4}=-(\frac{y}{3y+4}+\frac{4}{3y+4})=-(\frac{1}{4}\...
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84 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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53 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
44 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
41 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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63 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
30 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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1answer
151 views

Use Taylor expansion to determine the leading error term in a quadrature

I'm working on the last problem in an assignment, and need some guidance on what to actually start by doing. The question is asking me to use taylor expansion to determine the leading error term (...
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95 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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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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4k views

Third order term in Taylor Series

What is the third order term in the Taylor Series Expansion? I know it will just be third order partial derivatives but I want to know how is it expressed in a compact Matrix notation. For instance ...
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1answer
53 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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31 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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Why doesn't a Taylor series converge always?

The Taylor expansion itself can be derived from mean value theorems which themselves are valid over the entire domain of the function. Then why doesn't the Taylor series converge over the entire ...
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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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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} \frac{af(x-h)+...
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30 views

Taylor polynomials: Why does $R_{n,x_0}(x) = o(|x-x_0|^n)\implies \lim_{x\rightarrow x_0}R_{n,x_0}= 0$?

Given a remainder term $R_{n,x_0}(x)$ of a n-degree Taylor polynomial at $x_0$ $$f(x) = T_{n,x_0}f(x)+R_{n,x_0}f(x)$$ Why does $$R_{n,x_0}(x) = o(|x-x_0|^n)\implies lim_{x\rightarrow x_0}Rn_{n,x_0}= ...
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1answer
41 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 $\lim_{x\...
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3answers
67 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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1answer
55 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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How does one approximate $\cos(58^\circ)$ to four decimal places accuracy using Taylor's theorem?

When one needs to compute say $\cos (58^\circ)$ with an error of at most $10^{-4}$, how does one go about it? What is an appropriate centre of the Taylor expansion, and how does one determine the ...
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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
54 views

Taylor polynomial approximation - Interval of convergence

Find the Taylor polynomial of order $n$ of the cosine function around $x=0$. Then find the largest interval in which the sequence of polynomials $\{p_n\}$ converges to $f(x) = \cos(x)$. I am having ...
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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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1answer
26 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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55 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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30 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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1answer
35 views

$X\sim\mathcal N(0,1)$, Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd?

If $X\sim\mathcal N(0,1)$ Why is $\Phi_X^{(j)}(0)=0$ for $j$ odd ? ($\Phi_X^{(j)}(0):j^{th}$ derivative of the characteristic function of the r.v. $X$) We computed $\displaystyle\Phi_X(t)=e^{-\...
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1answer
20 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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40 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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109 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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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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1answer
46 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
44 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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29 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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42 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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26 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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84 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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34 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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33 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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31 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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43 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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185 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 $1$?