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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Taylor Series Expansion for $\tan x$

I'm trying to determine the Taylor series expansion for $\tan x$: I know that the $n$th derivative of the expansion must be the same as the $n$th derivative of the function. Please help, I have no ...
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1answer
19 views

Showing a function is identically zero using taylors theorem

Suppose we have a function $x(t)$ that is analytic everywhere, $x(t)\geq 0$, and $x(0)=0$. Suppose further that we know $x(t)$ is locally zero. ie. $x(t)=0$ for some small $t>0$. Is there a way ...
2
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1answer
91 views

How would I integrate $e^{e^x}$?

Is there a way to integrate: $e^{e^x}$ without using a Taylor or McLaurin Series expansion?
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1answer
50 views

Taylor series of a rational function

I am facing some complicated integral, which part of it is $$\frac{z^{M-1}}{(1+(\eta z)^n)^p}$$ I think if I find the taylor series of this part the integral might be solved. So, can someone help me ...
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1answer
88 views

Taylor Expansion of $ \frac{1}{x} $ about x = 0

I'm confused with the following problem: The expression $ \frac{1}{x} $ is clearly not defined at x = 0. However, I read that it could be expressed as a series using the idea $ ...
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1answer
34 views

Finding $\frac{\partial ^8 f}{\partial x^4\partial y^4}$

Given the function $f(x,y)=\frac{1}{1-xy}$ find the value of$\frac{\partial ^8 f}{\partial x^4\partial y^4}(0,0)$. First I developed the function into a taylor series using geometric series ...
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4answers
89 views

Taylor Series of $ \frac{1}{1-x^2} $ about x=2

I am trying to form a taylor series of the following: $ \frac{1}{1-x^2} $ about $x=2$ I tried factoring the equation such that it becomes the following: $ \frac{1}{{(1+x)}{(1-x)}} $ I tried to ...
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1answer
63 views

Does Taylor's theorem apply here?

Let $U\subset \mathbb{R}^n$ be open and $f:U\to \mathbb{R}^n$ with $x\in U$ and $\xi$ sufficiently small. Suppose that the following hold: $f(x+\xi)=\sum_{\alpha=0}^k ...
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3answers
269 views

Find a power series representation for the function $f(x)=\frac{(x-1)^2}{(3-x)^2}$

I tried to separate it and found the sum of $$\frac{1}{(1-x/3)^2}$$ but then I got stuck with having to multiply my sum with $(x-1)^2$ . I tried looking online but there's close to nothing about ...
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2answers
261 views

Express $(1-z)^{-1}$ as a power series around $z_0=-1+i$.

I need to express $(1-z)^{-1}$ as a power series in powers of $(z+1-i)$. I would like some guidance on the complex analogue of power series and in writing out this particular case. Many thanks for ...
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1answer
90 views

How to find the series $\sum_{n=1}^{\infty}\frac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$

Find this sum $$\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n}.\qquad (-1\le x\le 1)$$ My idea: let $$f(x)=\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$$ then we have ...
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3answers
196 views

Why Taylor series does not converge for all x in the domain of the function

Example: $$ f(x)=\frac{1}{1+x} \qquad x\neq-1 $$ $$ f(x)=1-x+x^2-x^3+x^4-x^5+\;... \qquad |x| < 1 $$ Why Taylor series does not converge for all x in the domain of the function?
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1answer
97 views

Prove inequality using taylor series

Let $0\leq p \leq 1$ and $\phi(t)=t\log \frac{t}{p} + (1-t)\log \frac{1-t}{1-p}$. Prove $\phi(t)\geq 2(t-p)^2$ for $t\in[0,1]$. Here's how I started. $$\phi'(t) = -\log ...
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0answers
110 views

Use Taylor Theorem and Taylor Expansion to Prove

Let $g: R\rightarrow R$ be a twice differentiable function satisfying $g(0)=1, g'(0)=0$ and $ g''(x)-g(x)=0$, for all $x$ in R Fix $x$ in R. Show that there exists $M>0$ such that for all natural ...
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2answers
251 views

Find the principal part of the Laurent expansion of $\frac{(z^{2}-2z-3)^{2}}{\cos(\pi z)+1}$ around $z_{0}=1$

Problem: The function $f(z) = \frac{(z^{2}-2z-3)^{2}}{\cos(\pi z)+1}$ has an isolated singularity at $z_0=1$. a) Find the principal (singular) part of the Laurent expansion of $f$ in a punctured ...
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1answer
164 views

Alternating series error bound

The taylor series for $ln(x)$, centered at $x=1$, is $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{(x-1)^n}{n} $$ Let $f$ be the function given by the sum of the first three nonzero terms of this series. The ...
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0answers
41 views

Finding f'(0) in a taylor series

While doing questions involving taylor series, I accidentally chanced upon an unorthodox, if more difficult way of solving for $f^{(n)}(0)$ of a given taylor series. I am wondering why it works, if ...
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2answers
90 views

Using Lagrange form of the remainder with cosh

I am trying to find "$\cosh 4$ using the sixth partial sum ($n=5$) of the Maclaurin series" for the function. I am also trying to use "the Lagrange form of the remainder to estimate the number of ...
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1answer
55 views

Generating Functions Interpretation - Expanding around other points?

Generating functions are incredibly useful for solving all kinds of combinatorial problems. Whenever they are used, though, the generating function is always expanded around $x=0$ to obtain the ...
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2answers
37 views

Generating a general term for a taylor polynomial

Let $f$ be the function given by $$ f\left(x\right) = \sin\left(5x +\frac{ \pi }{4}\right)$$ Let $P\left(x\right)$ represent Taylor polynomial of $f$ centred at $x =0$. Generate the general term for ...
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0answers
80 views

Taylor series expansion for $e^{\sin{x}}$ [duplicate]

Given the function $$f(x)= e^{\sin{x}}$$ I have to write it without using the exponential or sine function. I came to this point $$f(x) = \sum_{k=0}^{\infty} \frac{\sin^k{x}}{k!}$$ How can I get ...
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1answer
24 views

Reference of the expansion of square root polynomials

What is the reference of the formulation given below by Robert israel, please inform me.. Given an even-degree polynomial $$P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_0 = x^{2n} (a_{2n} + ...
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2answers
2k views

Power expansion for the square root of an even degree polynomial

I am reading an article from 1936 with something that looks like an easy way to solve Riccati equations with variable coefficients as nice polynomials.The link is : ...
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5answers
832 views

Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$

Prove that $$\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$$ The exercise provides the following hint: $\,\,\displaystyle{n \choose k}={n\choose n-k}$. Any help?
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19 views

How can i Taylor expand to get a difference approximation formula using

How can i Taylor expand to get a difference approximation formula using $y'''(0)=ay(-h)+by(0)+cy(h)+dy(2h)+O(h^p)$ where $O(h^p)$ needs to be as high as possible? i.e how can i Taylor expand the ...
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1answer
47 views

Lagrange Remainder and Intervals of convergence

(a) Determine the largest interval centered at $c=0$ on which we can be sure that $\lvert \cos(x) -(1-\frac{x^2}{2})\rvert < 10^{-6}$ (b) Let $T_n(x)$ denote the Taylor polynomial of order $n$ for ...
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1answer
60 views

Convergence question and degree of polynomial

I'm currently teaching myself power series and Taylor's theorem for complex analysis and I'm having trouble answering questions of the following form: $1)$ Suppose the power series ...
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2answers
60 views

What is series coefficient for $f(x)=\csc^2 x - \frac1{x^2}$?

What is general formula for Maclauren series expansion for $f(x)=\csc^2 x - \frac1{x^2}$ ?
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44 views

Why this function is elementary and its pair is not?

Why $$f(x)=\frac{2 \zeta (2)}{\pi ^2}+\frac{6 \zeta(4) x^2}{\pi^4}+\frac{10 \zeta(6) x^4}{\pi^6}+\frac{14 \zeta(8) x^6}{\pi^8}+\cdots$$ is elementary while $$g(x)=\frac{4 \zeta (3)x}{\pi ...
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2answers
31 views

Taylor Expansion of $\frac{x^4}{9+x^3}$ using elementary series

I have exhausted my book of tricks trying to do a series expansion of: $$f(x)=\frac{x^4}{9+x^3}$$ It is trivial to obtain by taking successive derivatives of the function, but I would like to know ...
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3answers
91 views

Proof using Taylor's theorem

Use Taylor's theorem to prove that $\displaystyle\lim_{n \to \infty} n \ln\left(1+\frac{1}{n}\right)=1$ I don't understand how to apply Taylor's theorem to a limit, especially one with a product of ...
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1answer
63 views

Taylors theorem application [duplicate]

I posted this question yesterday, and, despite getting answers, I am still confused how to solve it: Use Taylor's theorem to prove that $\displaystyle\lim_{n \to \infty} n ...
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0answers
47 views

Proving Taylor's Theorem by integrating n times?

From Arfken's Mathematical Methods for Physicists (7th ed.)... The remainder, $R_n$, is given by the n-fold integral ...
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0answers
155 views

How many terms to use in a Taylor series for local truncation error

So from my understanding for a finite difference approximation, you're supposed to expand the series "about" the point $x$, e.g., $$u(x+h) = u(x) + h \ u'(x) + ...
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3answers
135 views

Taylor series: $\sin x = x$?

Taylor series are used to expand a function to a series of functions that sometimes makes calculations easier. The more terms of a series we consider the more precise the solution would be. ...
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0answers
42 views

Manipulating Taylor expansion to contain sample mean, variance, skewness, and kurtosis

I have the following expression: $$\frac{1}{p} \ln\left(1+\frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n} \sum_{i=1}^n x_i^2 + \frac{p^3}{3!n} \sum_{i=1}^n x_i^3 + \frac{p^4}{4!n} \sum_{i=1}^n ...
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3answers
60 views

Taylor Polynomials question

Use Taylor polynomials at $x=x_0$ to approximate $\sqrt8$. I don't understand the point of Taylor polynomials here. If $T^{(0)}=f(x_0)=\sqrt8$, then what is the point of doing subsequent Taylor ...
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0answers
406 views

Infinitely differentiable function with divergent Taylor series?

I'd greatly appreciate it if someone could provide examples of the following: 1) A infinitely differentiable function whose Taylor series does not converge to the function. 2) An infinitely ...
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0answers
30 views

Taylor Expansion of a Summation

I am trying to get a first order Taylor approximation of the following expression: $$ \ln \left( \sum_{j=1}^{\infty} \pi_{a,j} \alpha_{a,j} \right) $$ around the mean value $\overline{\pi \alpha}$. ...
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2answers
131 views

If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...
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4answers
112 views

Maclaurin series of $\frac{1}{1+\sin x}$

Find the terms through degree four of the Maclaurin series of $f(x)$. $$f(x) = \frac{1}{1+\sin x}$$ My work: The Maclaurin series for $\sin x$ up to degree $4$ is $x - \frac{x^3}{6} + ...
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1answer
41 views

Expansions onto “bases”…?

When we consider expanding functions into fourier series, or taylor series, or onto the spherical harmonics-are these projections onto a basis? Are these bases complete? How can we show this? I know ...
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2answers
50 views

Taylor Polynom inequality

So the question is like this: Given $f(x)=\cos x$, find the taylor Polynomial of degree 2 and 4 and prove: $$P_2(x) < f(x) < P_4(x).$$ so I calculated these two polynomials: ...
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1answer
34 views

Approximation of $\sqrt{1+wi}$

How can $\sqrt{1+wi}$ be approximated? where $-\infty<w<\infty$; My aim here is getting rid of the square root. I've tried binomial, Maclurin and Taylor series around various points. but they ...
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1answer
44 views

What is the series expansion of $f(z)\cdot\exp\left({s\,\log(z)}\right)$?

For analytic $f$, how can I represent the expression $f(z)\cdot\exp\left({s\,\log(z)}\right)$, i.e. $f(z)\cdot z^s$ in the form $$\sum_{n}^\infty\left(\sum_{k}^\infty a_k s^k\right)z^n,$$ at least ...
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0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
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39 views

Question about asympotic expansions!! please help!

Question: Find the constants $$a_0, a_1, a_2$$ in the asympotic expansion $$\int_0^x t\sqrt{ln(t)} dt$$ = $a_0(x^2)(lnx)^\frac 12$ + $a_1\frac {x^2}{(lnx)^\frac 12}$ + $a_2\frac {x^2}{(lnx)^\frac ...
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2answers
98 views

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

I have been given the function $$F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,{\rm d}t$$ for $x\ne 0,$ $F(0)=\frac12,$ and charged with finding a Taylor polynomial for $F(x)$ differing from $F$ by no ...
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1answer
38 views

A power approximation function

I am trying to construct a function that would approximate $a^b$ using Maclaurin series. Here are my reasoning: Since $$a^b=e^{b\ln a}$$ and $$e^x=\sum^{\infty}_{k=0} \frac{x^k}{k!}$$ it should ...
4
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2answers
314 views

Taylor series of the inverse of $x^4+x$

I would like to expand the inverse function of $$g(x) := x^4+x $$ in a taylor series at the point x = 0. I calculated the first and second derivate at x = 0 with the rule of the derivation of an ...