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: $\sin x = x$?

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
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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 ...
0
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3answers
56 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
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0answers
372 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
29 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
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. ...
2
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4answers
110 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
40 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
48 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
33 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
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1answer
43 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
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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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0answers
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
96 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
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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
votes
2answers
278 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 ...
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0answers
38 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: ...
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3answers
128 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 ...
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1answer
108 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
84 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: $$ ...
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2answers
78 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
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1answer
74 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} ...
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2answers
39 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.
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1answer
38 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 ...
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0answers
54 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 ...
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1answer
116 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
152 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 $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
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1answer
61 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
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1answer
57 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 ...
0
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3answers
53 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?
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6answers
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Don't understand why this binomial expansion is not valid for x > 1

today I'm studying binomial expansion and I'm a little confused about when certain expressions are valid. E.g. take this solution from my textbook: I understand that $(1-x)^{-1}$ has an infinite ...
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1answer
91 views

Getting the exact value of the Maclaurin series for $\sinh$

I am new to Maclaurin series and I am trying to find the exact value for $\sinh(x)$. Let say I have the highest power of $5$ and value of $x$ is $2$. How do I start?? Sorry I am really new to this.
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0answers
50 views

Multivariate Taylor Polynomial

The Exercise: Calculate the Taylor polynomial of degree 3 of $f(x,y,z)=x^5y^4z^3$ at $(1,1,1)$ in an arbitrary direction $h$. Use Taylor's theorem to get a bound on the remainder when using this ...
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2answers
132 views

Taylor series remainder (Lagrange)

I have this function $f(x) = ln(1+x)$ and I want to come up with the Maclaurin series for it up to $n = 3, a = 0$. I calculate that the remainder $n=3$ is $R_3(x)=-\frac{6}{4!(1+c)^4}x^4, \; c \in ...
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1answer
30 views

$\frac{P(x)}{(ax+b)^n }= \frac{c_1}{ax+b }+\frac{c_2}{(ax+b)^2 }+…+\frac{c_n}{(ax+b)^n }$

Let $n\geq 1$ be an integer, $P(x)$ be a polynomial of degree lower than $n$. Prove, if $a$ is real and $a \neq 0$ then: $$\frac{P(x)}{(ax+b)^n} = \frac{c_1}{ax+b} + \frac{c_2}{(ax+b)^2} + ...
0
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1answer
25 views

Functions and their Taylor polynomials

Given a function $f$ from $\Bbb{R}$ to $\Bbb{R}$, we define $P_{f,n,a}$ to be the Taylor polynomial of $f$ of degree $n$ at $a$ (if the function itself is clear from the context, we simply write the ...
3
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1answer
69 views

Showing $|a_k | \le 1$

Let $A$ be the closed unit disk $A= \{z \in \mathbb{C}: |z| \le 1 \}$. Suppose $f$ is an entire function whose Taylor series centered at the origin is $$\sum_{k=0}^{\infty} a_kz^k$$ and that $f$ maps ...
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1answer
39 views

Taylor & MacLaurin series

I have a problem that I can not understand at all. I know how to calculate taylor/macLaurin for $\cos x$, $\sin x$ and $e^x$ etc. However when I have for example: $\sin x \cos x$ or $\sin x + \cos ...
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2answers
47 views

How to prove a series is greater than zero over an interval?

Show that the series $\sum\limits_{k=0}^\infty \frac{(-1)^k(x^{2k+1})}{(2k + 1)!}$ is greater than zero for $0<x\leq \sqrt{6}$ For a function to show something was greater than zero over an ...
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2answers
87 views

Remainder of Taylor series

The Taylor series of the function $$f(x) = \int_{1}^{\sqrt{x}} \ln(xy)+ y^{3x} dy + e^{2x}$$ at the point $x = 1$ is $$e^2 + (x-1)\left(2e^2+\frac{1}{2}\right) + ...
3
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0answers
151 views

Loss of Significance problems - Taylor Expansion

(2) This question addresses the notion of loss of significance. You are encouraged to revisit the Taylor series expansion that you have learned in calculus, as you will need to apply it here. Explain ...
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0answers
79 views

Estimate probability of event using moments of a distribution or a Taylor expansion involving the moments

Let's say we have four moments $(\mu_1, \mu_2, \mu_3, \mu_4)$ of a probabilty distribution of a random variable $X$ and the goal is to get the probability $\rm{P}(X \leq t)$ for a certain value of ...
2
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2answers
7k views

What's the Maclaurin series for $\arcsin(x)$?

I solved the problem by using a known series: $\frac{1}{\sqrt{1-x^2}}$, but the solution I got is wrong. Also, I'm not sure what to do with the constant of integration $C$. Where is my mistake? $$ ...
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1answer
67 views

Taylor series about (3+x)/(x+4)*exp(-x) expanded at x = - 4. How do i replicate what I see in wolfram?

I'm puzzled by wolfram alpha's results. If i ask 'series $\frac{3+x}{4+x} (\exp (-x))$ expanded at -4' It will return a series. What method does it use to do this? I'm familiar with Taylor ...
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0answers
33 views

Multivariate taylor expansion application

I have to show that, given the operator P such that: $P f(x) = \sum_{|\alpha| \leq m} c_{\alpha} \frac{d^{\alpha} f}{dx^\alpha} = 0$ (Using multivariable multi-index notation) Then the same would ...
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0answers
130 views

Convergence of Taylor series of analytic function

Let $f(x)$ be analytic on $D= \{x \in \mathbb R^2: |x|< 1\}$. Then for $x_0 \in D$ there is an open set $U$ such that for all $x \in U$: $\sum_{n=0}^\infty a_n (x-x_0)^n = f(x)$, that is, the ...
2
votes
2answers
123 views

Evaluate the sum of the series

Was given the following infinite sum in class as a question* (while we were talking about taylor series expansions of $\ln(1+x)$ and $\arctan(x)$: $$1 + \frac12 - \frac23 + \frac14 + \frac15 - \frac26 ...
0
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1answer
55 views

taylor series for cosx around 0

Hey, I have the following limit, and I would like to know if it's possible to use the maclaurin series for cos(x) around 0. Is it okey to do the step I have done in the picture bellow? and let's say ...
0
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1answer
38 views

Taylor expansion for $\frac{\sin(x)}{x}$ at $x=0$?

I don't understand how to expand this function, for example how do I plug $x=0$ into $\frac{\sin(x)}{x}$ ? All derivatives of the function have $x$ in the denominator, I'm just confused. Thanks for ...
2
votes
1answer
130 views

Numerical Approximation

we aim to find an approximation of exp(x) a) determine N an integer and g \in (-log(2)/2, log(2)/2] such that x=N*log(2) + g b) Think of a way to use the Taylor expansion with four terms of exp(x) ...