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.

learn more… | top users | synonyms (1)

5
votes
1answer
45 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 ...
2
votes
2answers
28 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 ...
3
votes
0answers
79 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 ...
0
votes
1answer
17 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} + ...
0
votes
2answers
1k 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 : ...
0
votes
5answers
150 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?
8
votes
1answer
130 views

Taylor Series of $\frac{1}{1-\cos x}$

The problem is, as the title suggests, to find the Power Series Expansion of $\frac{1}{1- \cos x}$ around $x=c$. What I've tried: Direct Computation: Derivatives get very ugly quickly, and don't ...
0
votes
0answers
14 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 ...
0
votes
1answer
20 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 ...
1
vote
1answer
38 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 ...
0
votes
2answers
48 views

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

What is general formula for Maclauren series expansion for $f(x)=\csc^2 x - \frac1{x^2}$ ?
0
votes
0answers
40 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 ...
0
votes
2answers
28 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 ...
-1
votes
3answers
67 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 ...
-2
votes
1answer
52 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 ...
0
votes
0answers
25 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 ...
0
votes
0answers
27 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) + ...
4
votes
3answers
98 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. ...
0
votes
0answers
23 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 ...
0
votes
3answers
33 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 ...
2
votes
0answers
80 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 ...
0
votes
0answers
27 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 $\overline{\pi \alpha}$. ...
3
votes
2answers
73 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. ...
2
votes
4answers
53 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} + ...
0
votes
1answer
29 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 ...
0
votes
2answers
38 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: ...
0
votes
1answer
30 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 ...
0
votes
1answer
35 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 ...
4
votes
0answers
86 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 ...
1
vote
0answers
36 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 ...
1
vote
2answers
67 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 ...
0
votes
1answer
36 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
votes
2answers
177 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 ...
1
vote
0answers
34 views

Find the right degree of the Maclaurin polynomial of $e^x$

Here is my question: What degree Maclaurin polynomial of $e^x$ must be taken to guarantee an estimate of $e$ to within $1 \times 10^{-6}?$ I know that the error term is: ...
1
vote
3answers
66 views

How is the Logarithm derived from the exponential function? (aren't they inverses?)

I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series ...
0
votes
1answer
57 views

taylor series expansion for a rational function

What is the Taylor Series Expansion (function of z ) for where $\eta$, $n$ and $p$ are positive real constants Based on the answers in the comments, does this mean that the taylor series is given ...
5
votes
4answers
79 views

Multiplying the long polynomials for $e^x$ and $e^y$ does not give me the long polynomial for $e^{x+y}$

As an alternative to normal rules for powers giving $e^xe^y=e^{(x+y)}$ I am multiplying the long polynomial of the taylor series of $e^x$ and $e^y$. I only take the first three terms: $$ ...
0
votes
2answers
63 views

Equality of a function and Taylor Series

Does the following function have a Taylor series of the form given below: $$\frac{1}{(1+(\eta z)^n)^p} = ...
3
votes
1answer
55 views

How to show that $e^{x+y} = e^x e^y$ by series expansion [duplicate]

I know that $e^xe^y=e^{x+y}$ but I want to show it by expanding the exponentials in MacLaurin Series. $$ \left(\sum_{n=0}^{\infty} \frac{x^n}{n!}\right) \left(\sum_{m=0}^{\infty} ...
0
votes
2answers
33 views

How can I see $\frac{1}{1 - e^{-u}} = \sum_{k=0}^{\infty} e^{-ku}$?

How can I see $$\frac{1}{1 - e^{-u}} = \sum_{k=0}^{\infty} e^{-ku} ?$$ I know it's related to Taylor series, but I don't get it.
0
votes
0answers
29 views

Taylor series of some stange function

Can somebody help find the Taylor series for: where p and n are real positive (not necessary integers)? Does it converge fast?
0
votes
1answer
33 views

Help with Taylor series problem

I am using maple to plot the graphs of e^e^x versus its truncated Taylor series around 0. For small values of x, the two graphs converge nicely, but once x<-3, my Taylor series loses control. Here ...
1
vote
0answers
43 views

Error Term of the Taylor series of cosh

I have the Taylor series of cosh $$\sum_{n=0}^\infty \frac {x^{2n}} {(2n)!}$$ and I know that this series converges for all x, but now I want to know if the series represents the function, in other ...
1
vote
1answer
93 views

Taylor series of $\frac 1 {1+x^2}$

I have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to construct ...
3
votes
2answers
123 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
1
vote
1answer
38 views

Should I use Taylor Expansion or the expansion of $e^x$ to express a second order differential (in this case)

I've been given this equation: $(1+x^2)\dfrac{d^2y}{dx^2} + 4x\dfrac{dy}{dx} + 2y = 0$ I've also been told that: $y=1, \dfrac{dy}{dx} = 1$, at $x=-1$ I've been asked to find a series solution of ...
2
votes
1answer
39 views

Taylor expansion of polynomial

Intuitively, I would expect the Taylor expansion around $x_0$ of a polynomial in $(x-x_0)$ to be identical to the polynomial. However, I cannot seem to show that/whether this is the case: For a ...
1
vote
2answers
34 views

Taylor Polynomial converges to the original function?

If $P_n(x)=x-\frac{x^2}{2}+\frac{x^3}{3}-..+\frac{x^{2n+1}}{2n+1}$ (It's taylor series of $ln(1+x)$ near x=0. Then can I say that: $\lim_{n\to\infty}{P_n(x)}=ln(1+x)$, please explain why or why not.
0
votes
3answers
49 views

A simple series

I don't do math a long time, so I completely don't remember how to prove that: $$ \sum_{i=1}^\infty \frac{i}{2^i} = 2 $$ Can anybody help me?
19
votes
12answers
11k views

What are the practical applications of the Taylor Series?

I started learning about the Taylor Series in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series. Question: ...