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 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
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4answers
86 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
80 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
78 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
40 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
40 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
56 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
121 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 ...
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2answers
156 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
62 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 ...
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1answer
59 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 ...
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3answers
54 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
3k views

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
96 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
53 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
145 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
31 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} + ...
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1answer
26 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 ...
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1answer
70 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
40 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
48 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
89 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) + ...
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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
84 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 ...
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2answers
8k 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
68 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
36 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
131 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 ...
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2answers
125 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 ...
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1answer
58 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 ...
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1answer
39 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
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1answer
134 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) ...
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2answers
61 views

Taylor expansion for Si(x)?

I want to find out what the Taylor expansion of $$F(x) = \int_0^x \frac{\sin(t)}{t} dt .$$ Am I wrong in saying that by the fundamental theorem of calculus, $F'(x) = sin(t)/t$? Should I continue ...
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2answers
122 views

Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$

Prove the following inequality: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ This should be solved using Taylor series. I tried expanding the left to the 5th degree and the right site to ...
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2answers
61 views

Maclaurin series for $f(x)=\frac{1}{1+x+x^2} $

What is the Maclaurin expansion of $f(x)=\dfrac{1}{1+x+x^2} $? Thank you! Edit: By multiplying both terms with $ (1-x) $ I got to $\dfrac{1}{1-x^3}-\dfrac{x}{1-x^3}$. Is it correct to transform ...
2
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3answers
95 views

$\lim_{x\to\infty} \frac{5\cdot5^x+3^x-4^x}{5^x +2^x+27\cdot9^x}$

How can I solve this limit. (Here $x$ belongs to natural numbers $\Bbb{N}$.) $$ \lim_{x\to\infty} \dfrac{5\cdot5^x+3^x-4^x}{5^x +2^x+27\cdot9^x}$$ My try: I tried using L'Hospital, expansions of ...
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3answers
73 views

Taylor expansion - what order would be preferred?

Let say you want to calculate the following limit: $$\mathop {\lim }\limits_{x \to 0} \left( {\frac{1}{{1 - \cos x}}\ln \left( {\frac{{\sin x}}{x}} \right)} \right)$$ Obviously, Taylor Expansion ...
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1answer
381 views

What's wrong in my Taylor series implementation in MATLAB?

I'm trying to code Taylor summation for a function in Matlab, I actually evaluate McLaurin making $x_0=0$, named a in this code after this notation: This is the code I've tried out so far: ...
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5answers
94 views

What does $a$ mean in Taylor series formula?

I'm trying to code the Taylor summation in MATLAB, being Taylor's formula the following: I've also seen $a$ denoted as $x_0$ in distinct bibliography. Problem is that I'm not sure how should I ...
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4answers
1k views

Using Taylor's Theorem to show that $\ln(1 + x^2) \leq x^2$

Can we show that if $\operatorname{abs}(x) \lt 1$, then $$\ln(1+x^2) \leq x^2\;,$$ using Taylor's Theorem? I am thinking of expanding it about $x=0$ but I got something like $$f(x) = -x^2 + ...
3
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2answers
541 views

Express $\sin nx$ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively

What are the expansions of $\sin nx $ and $\cos nx$ in terms of $\sin x$ and $\cos x$ respectively? (here $n \in \mathbb N$). Maybe this is solved problem or there is new technique to ...
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1answer
37 views

Tangent and Taylor polynomials

We know that this series $x+ \frac{x^3}{3}+\frac{2x^5}{15}+\ldots$ is convergent in $|x|\lt \pi/2$, furthermore it converges to $\tan(x)$. I would like to know if we restrict to finite terms of this ...
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1answer
44 views

Asymptotic Expansion in zero of $\frac{1}{\ln(1+x)}$

On wolfram the expansion is: $$\frac {1}{x} + \dfrac{1}{2} ...\,.$$ But I don't understand from where it outside comes the $\frac{1}{2}$ thanks
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0answers
185 views

taylor series of $\sin t/t$

I know that the taylor series expansion around zero of $\sin t/t$ is: $\sum_{k=0}^{\infty} (-1)^k \frac{(t^{2k})}{(2k+1)!}$ , I need to find its radius of convergence. I saw a few solutions that claim ...
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2answers
39 views

Maclaurin Expansion of $\frac{x}{\sqrt{4-2x}}$

Maclaurin Expansion of $\frac{x}{\sqrt{4-2x}}$ up to order 4. I really don't know how to do this, I can't find a helpful Maclaurin Series in my formula book to help me. I want to do $x(4-2x)^{-1/2}$ ...
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0answers
42 views

Landau identification

Calculate taylor series for $x\rightarrow +\infty$ at the higher order allowed by the approximation present in it. $$ \sqrt{x^6+x^5-2x^3+O\left(x^2\right)} $$ I made this: $$ ...
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2answers
36 views

Tayors series exansion of $(1 - x )^n$ where $0<x<1$ and $n \ge 0$

I want to find Taylor's series or Maclaurin's series expansion of the following. $$(1 - x)^n \ \text{ where }\ \ 0 < x < 1 \text{ and }\ n \ge 0$$ will it be same as that for $$(1 + x)^n ...
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1answer
31 views

Expanding One Function in Powers of Another

One sees here that it is possible to expand $f(x) = 2x^3 + 7x^2 + x - 6$ in powers of $x - 2$ by taylor expanding $f(x) = f(x - 2 + 2) = f(2 + h)$ about $2$, and this idea can be used in deriving the ...
12
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4answers
816 views

Taylor expansion of $(1+x)^α$ to binomial series – why does the remainder term converge?

For $α ∈ ℝ$ the function $g_α \colon B_1(0) → ℝ, x ↦ (1+x)^α$ is $C^∞$ and $g_α^{(n)}(x) = n! \tbinom{α}{n}(1+x)^{α-n}$, where $\tbinom{α}{n} = \frac{α(α-1)\cdots(α-n+1)}{n!}$ is the generalized ...