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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Evaluate an integral $\int_{\gamma } z^{n}e^{\frac{1}{z}}dz$ maybe laurent series or taylor expansion?

Please help me with this one guys, I am stuck like a truck trying to get out of thick mud. Evaluate: $\int_{\gamma } z^{n}e^{\frac{1}{z}}dz$ $\gamma$ is the circle f radius 1 centered at 0 and ...
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37 views

Laurents Series Expansion Complex Analysis

So here is the problem, I am having a lot of trouble with laurents expansions and if you guys even know any sources where I can learn these really well and very simply then that would be a great help. ...
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27 views

Remainder in taylor formula

I found on a book a version of Taylor's formula like this: ...
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17 views

proof of property of exponential distribution, using taylor polynomial

I want to prove that if we have an exponential distribution with parameter $\lambda$, we have that $P(X \le x)=\lambda x + o(x)$. I want to do this by using Taylor-series and the lagrange remainder ...
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Find the order of a function.

Consider the function $(x + 2)\cos^2 x$. Determine its order in terms of big-O notation. (A) $O(x)$ (B) $O(x^2)$ (C) $O(\log (x))$ (D) None of the above
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Take 2: When/Why are these equal?

This didn't go right the first time, so I'm going to drastically rephrase the query. As per this previous question, I am wondering if the two series ...
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92 views

definition of the constant $e$

To my knowledge there are two possible ways to define $e^x$ $$e^x = \sum_{i=0}^{\infty}\frac{x^i}{i!}$$ $$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n$$ So my question is: Why does… ...
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Approximation of $x\log(x/a)$ for $x$ near a

I'm trying to see where the approximation $$(x-a) + ((x-a)^2)/2a$$ of $x\log(x/a)$ comes from (for x near a). Might be missing something very trivial but I've already tried the usual expansions ...
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43 views

Taylor error approximation

So a Taylor polynomial is given by the Taylor formula, but how do I approximate the error? I see on wikipedia: $$R_k = \frac{f^{(k+1)}(s)}{(k+1)!} (x-a)^{k+1}$$ Do I just pick any $s$ between $x$ ...
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Taylor series argument help for details

http://arxiv.org/pdf/math/0601086.pdf I want to ask about the Taylor series argument on page 25 above. The proof want to show $e(x)<e(x_0)$ where $e(x)=c(x,y)-c(x,y_0)$ The problems are: I ...
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63 views

Taylor expansion for a multivariable function

\begin{align} T(x_1,\dots,x_d) &= \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty \cdots \sum_{n_d = 0}^\infty \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + ...
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1answer
33 views

Taylor series convergence with natural logs

I am working on this problem. Find Taylor series of function $f(x)=\ln(x)$ at $a = 6$. $$f(x) =\sum_{n=0}^\infty c_n (x- 6)^n$$ I seem to be having trouble with the interval of convergence can ...
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Taylor series of an integral function

Problem $$I(x) = \int_{1}^x \frac{e^t - 1}{t}$$ Find $I'( \sqrt{x} )$. Solution We know that $F'(x) = f(x)$ by the fundamental theorem of calculus so $$I'(x) = \frac{e^t -1}{t}$$ And so $$I'( ...
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A Taylor series question

The Taylor series for $cos(x)$ about $x=0$ is $1-x^2/(2!)+x^4/(4!)-x^6/(6!)+...$ If $h$ is a function such that $h'(x) = cos(x^3)$, then the coefficient of $x^7$ in the Taylor series for $h(x)$ about ...
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1answer
23 views

Taylor Series & complex analysis

I am taking complex analysis. There's a question in the book when trying to prove the theorem, and the theorem goes like this: If $f$ is analytic in the disk $|z-z_0|<R$,then the taylor ...
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44 views

Taylor's approximation

Our lecturer once showed us that it is possible to approximate the value of 'e' with Taylor's approximation of order - whatever, lets say 3. How would. The result was something like this: $$ ...
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Question about Taylor's series

Is there an example of a function whose taylor series converge at every point but does not equal the value of the function at every point?
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50 views

High Order Derivative Using Maclaurin Series

Use the Maclaurin series to solve the following: $$ \frac{d^6}{dx^6}(x^4e^{x^2}) $$ I got about halfway through the problem before getting stuck. I am not sure how to solve it... Any advice? Also, ...
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Big-O Notation for remainder terms in Taylor expansion

The Big-O notation is commonly used in Taylor expansions of the form $$f(x+\epsilon)=f(x)+\epsilon f'(x)+O(\epsilon^2)$$ to say that the remainder term grows at least quadratic around $\epsilon=0$. ...
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1answer
46 views

Approximation of $\frac{1}{2^{N}} \binom{N}{\frac{1}{2}(N + s)} $ for big N

I have the following function that I have to approximate for big N and I really have no clue what to do - I hope that anyone can help me out: $$P_N(s) = \begin{cases} \frac{1}{2^{N}} ...
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Taylor series for small complex number

Show from Taylor's formula that in first order in a small complex number $z=x+iy$ we have following approximation: $|1+z| \approx 1+x$ and $\frac{1}{|1+z|^3} \approx 1-3x$. If I define a function ...
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Question about Big O notation for asymptotic behavior in convergent power series

Examples of such use of Big O notation can be found for instance on Wolfram Alpha here. More details on the Wikipedia page. The idea, as I understand it, is that the term between parenthesis in Big O ...
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35 views

Converse of Taylor series expansion

I am doing Taylor series expansion. A/c to my professor an infinitely diffrentiable function w.r.t to some variable can be represented as as a polynomial in that variable of $n$ degree where $n$ ...
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40 views

Using Taylor expansion to evaluate infinite sum

How do I use the Taylor expansion of $$(1+x)^{-\frac{1}{2}} $$ to evaluate $$ \sum_{n=0}^{\infty}\binom{2n}{n}\left(-\dfrac{6}{25}\right)^{n} $$ Thanks
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Taylor Series Calc 2

I am not sure how to find a series representation for the natural log. If anyone can show me some helpful steps to solve this problem it would be greatly appreciated. What is the Maclaurin series ...
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1answer
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Help with Taylor Series

I am trying to find a Taylor series for the following function: ${1\over 1-9x}$ centered at c = 7 I browsed through my Calc II book and found that I can use the general formula for a Taylor series ...
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21 views

Bounding the quotient of random variables

I have two non-negative random variables $X, Y$ with finite expected values and variances, and I want to bound $E(X/Y)$ from above. I was reading these notes and they do a two-variable version of ...
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51 views

Expanding $\ln(1+f(x))$ around $f(x)=0$ when we do not know whether there is an $x$ such that $f(x)=0$.

I want to expand $\ln(1+f_T(x,\theta))$ about $1+f_T(x,\theta)=1$. What I have in mind is something like $$ \ln(1+f_T(x,\theta))=\ln(1)+f_T(x,\theta)-\frac{1}{2} \frac{1}{1+\tilde{f}} ...
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Taylor expansion, integration by parts, and the integration of dt.

So my notes say, for a continuous function we have $$ \int_a^x f'(t)dt = f(x) - f(a) \tag 1 $$ which I understand. So re-arranging gives. $$ f(x) = f(a) + \int_a^x f'(t)dt \tag 2 $$ or $$ f(x) ...
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Why is the Taylor expansion of $\cos$ decreasing?

Why is the Taylor expansion of $\cos$ decreasing ? $\cos(t)=1-\frac{t^2}{2!}+\frac{t^4}{4!}-\frac{t^6}{6!}+...$ such that one can estimate $\cos(t)<1-\frac{t^2}{2!}+\frac{t^4}{4!}$ I ...
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Laurent Series and Taylor Expansion of $ 1 / (e^z - 1) $

Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."? I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, ...
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25 views

Taylor Polynomial Question

Can anyone help me with this question? Calculate the Taylor polynomials $T_2(x)$ and $T_3(x)$ centered at $x=\frac{\pi}{6}$ for $f(x)=\sin(x)$.
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Maclaurin Series for a natural logarithm

Can anyone please help me with this question? Find the Maclaurin series and the interval of convergence for $f(x) = \ln(1-7x^9)$ I thought the answer was $$\sum_{n=1}^{\infty} (-1)^n ...
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31 views

If $y = e^{x(y-1)}$ then $y \approx 1-2(x-1)$ when $0<x-1<<1$

Assume $y = e^{x(y-1)}$. Then $y \approx 1-2(x-1)$ when $0<x-1<<1$ I thought of something like that: $$ e^{x(y-1)} = e^{-2(x-1)}e^{xy+x-2}=(1-2(x-1)+O(x-1)^2)e^{xy+x-2}$$ But I failed ...
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30 views

Taylor Series Maclaurin Series Interval Expansion

Hi! I am currently woking on some clack online homework problem. I really have no idea how to approach this problem. If someone could help me solve this question I would greatly appreciate it!
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22 views

Taylor series and Maclaurin Series expansion

Hi! I am currently working on some calc2 online homework problems on Taylor series and Maclaurin series. I have tried a few different answers to this question, but I am really not sure how to go ...
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3answers
33 views

Expand a sin(x^3) in Maclaurin's series and find a 30th derivative at (0)

I have a big task and problems with it. I have to expand this function in Maclaurin's series. $$cos(x^{3})$$ I tried expand it as $$\sum_{n=0}^\infty (x^n)/n!$$ but it's for $cos(x)$. So i don't know ...
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proof that Even powers of an odd function's taylor polynomial vanish

Let $f$ be a $k$ times continuously differentiable function defined on a neighborhood of $0 \in \mathbb{R}$. Show that if $f(-x) = -f(x) \forall x \in \mathbb{R}^n$, then the coefficients of the ...
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23 views

For which $x$ does the 1st Taylor polynomial for ln(1+x) give 2 decimal places accuracy?

My work: To find the polynomial approximation: ln(1+0) = ln(1) = 0 so the constant term is 0. $\frac {d}{dx} \ln(1+x) = \frac {1}{1+x}$, and at $x=0$, this is equal to 1. So the polynomial we're ...
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Expand a function in Maclaurin's series

Please help me expand this function in Maclaurin's series to find an interval of convergence. I tried to turn it into this expression: but it doesn't give me the result. Help me please it's ...
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28 views

Radius of convergence of $(1+x)^p$

Problem: Show that $(1+x)^p$ converges everywhere for $p \in \mathbb{N}$, and for $|x| < 1$ otherwise. My work: I think that if $p \in \mathbb{N}$ then the Taylor series will just be a polynomial ...
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62 views

Decide convergence of the series

Using Taylor expansion decide convergence of the series: $$\sum_{n=1}^{\infty}(e-(1+{{1}\over{n}})^n)^p = \sum_{n=1}^{\infty}a_n$$ I expanded $a_n$ like this $a_n = (e-(1+{{1}\over{n}})^n)^p = ...
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31 views

Partial derivatives + Taylor's Formula in several variables

Given a function $f(x) = (x_1+...+x_n)^k$, how do we show that $$D_1^{j_1}\cdots D_n^{j_n}f(x) = k!$$ if $j_1+...+j_n = k$?
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Find the Taylor series of $f(x) = e ^{- 1 / x^2}$

Find the Taylor series about 0, the function defined as: $f(x) = e ^{- 1 / x^2}$ if $x \ne 0$ and $f(x) = 0$ if $x=0$ and What can i conclude of the resulting? First i note that the function f is ...
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Backward Difference Formula: Solving for First Derivative with a Limited Set of Knowns

I am trying to solve for $f''(x)$ by using only the following in the set ${ f(x_0), f((x_0-h), f(x_0 +h)}$. I realize that I am suppose to use Taylor's Theorem. This should help with the cancellation, ...
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41 views

Taylor expansion at infinity and optimal value at infinity of a function.

Given a function $f(x)$ that we need to minimize on the hold space, i.e., $$\mbox{minimize} \;\; f(x):\;\;\; \mbox{subject to }\; x\in X.$$ Suppose this function is bounded, i.e., $|f(x)|\leq \gamma$ ...
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29 views

Third degree Taylor polynomial in two variables

How does one find the third-degree Taylor polynomial of $f(x,y) = (x+y)^3$ at the points $(0,0)$ and $(1,1)$? Many thanks
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24 views

Solution of an equation with a taylor expansion

Show that $$x=ne^{-x}$$ with $n \in \mathbb{N}$ and $n\ge1$ has one solution $x_n$ such that: $$x_n = \log(n)-\log(\log(n)) + o(\log(n))= \log(n)-\log(\log(n)) + o(1)$$ My try: Let be ...
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20 views

Calculating MacLaurin series for $\frac{1}{1-x^2}$

We have the M-series for $\frac{1}{1-x} = \sum_{n=0}^\infty x^n, \frac{1}{1+x} = \sum_{n=0}^\infty (-x)^n,$ and $\frac{1}{1-x^2} = \sum_{n=0}^\infty (x^2)^n$. I need to use the product of the first ...
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22 views

Inequality doubt with taylor expansion

Can I prove that $\forall x>0$ $$e^{x/(1+x)} < 1+x$$ Showing that $e^{x/(1+x)} = 1+x-\frac{x^2}{2}+o(x^2)$ and so $-\frac{x^2}{2}+o(x^2)<0$ for all $x>0$? How i can be sure that $o(x^2)$ ...