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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1answer
111 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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1answer
59 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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2answers
63 views

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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3answers
82 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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3answers
104 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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2answers
187 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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1answer
103 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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1answer
51 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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120 views

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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32 views

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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2answers
56 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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27 views

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

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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1answer
2k views

Taylor Series of Hyperbolic Cotangent Coth(x)

Expanding about 0 gets me a divergence on the first term, and the wikipedia article says nothing about how to derive it other than taylor series. It makes me think I'm supposed to use Laurent Series, ...
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1answer
39 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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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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2answers
44 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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3answers
59 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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37 views

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
39 views

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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54 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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147 views

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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1answer
56 views

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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382 views

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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1answer
40 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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37 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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1answer
39 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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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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3answers
66 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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1answer
33 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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170 views

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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88 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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143 views

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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1answer
39 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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60 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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51 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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359 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
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6answers
4k views

Intuition explanation of taylor expansion?

Could you provide a geometric explanation of taylor expansion?
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33 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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1answer
3k views

Difference between the Laurent and Taylor Series.

I have never taken complex analysis, but I am preparing for a GRE for this week end and I am trying to learn a bit about Laurent Series. So far what I get is that the Laurent Series are of form ...
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1answer
23 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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2answers
30 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 $o(x^2)$ ...
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62 views

Taylor expansion of $\frac{1}{2-z-z^2}$

The problem is: Find the Taylor expansion of $f(z):= \dfrac{1}{2-z-z^2}$ on the disc $|z| < 1$ So far I have used partial fractions to obtain $f(z) = \dfrac{1}{3}\left(\dfrac{1}{1-z} + ...
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General 2D taylor surfaces from axial behaviour and discrete points

I have a problem as follows: I have a nonlinear function, f(x,y), for which I (numerically) know the axial behaviours, f(x,y0) and f(x0,y), where x0 and y0 are constants. I can calculate discrete ...
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169 views

Find the power series for $f(x) = \frac{\cos(x^3)}{2x^2}$

I'm pretty sure if it were just $\cos(x^3)$ i could subsititue $x^3$ for $x$, everywhere in the known series, but what do I do because it's divided by $2x^2$?
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2answers
444 views

Prove Euler's Formula using MacLaurin Series

How can you prove $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ (Euler's Formula) using MacLaurin Series? Thanks!
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1answer
43 views

Taylor expansion with non integer exponents in the rest

Consider the function: $$f(x)=\sqrt[3]{8x^2+4x+1}$$ 1) Find $a,b,\alpha,\beta$ such that: $$f(x)=ax^\alpha+bx^\beta+o(x^{-1/3})$$ 2) Find $A=f([0,+∞[)$ and prove that $f:[0,+∞[\rightarrow A$ is ...
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2answers
116 views

Taylor Series looks like exponential

guys! I need a little help. I have this series $$ \sum_{k \ge 0} \frac{\Gamma(j)}{\Gamma(j+k/2)}(-t)^k $$ where $j \in \mathbb{N}$. I need to know if the limit of this function when $t$ goes to ...
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1answer
53 views

MacLaurin powerseries and interval of convergence

Given the function $f(x) = 5/(6*x^2-x-1)$, (a) Expand into MacLaurin powerseries the function $f$ up to order $3$. (b) Find the interval of convergence of it. (a) I will use the type of ...