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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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
56 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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59 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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231 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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179 views

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
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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219 views

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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62 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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38 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
41 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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1answer
175 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
57 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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408 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
43 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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70 views

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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2answers
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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45 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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1answer
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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
69 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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213 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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1answer
34 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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21 views

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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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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2answers
89 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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1answer
40 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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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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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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52 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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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
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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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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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63 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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19 views

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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177 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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Is there a Taylor series for vector cross product?

I have this equation, where $u,v,w,a,b,Ɵ$ are constants. The RHS comes from the Geometric definition of the LHS $(u,v,w)(a,b,c)=||(u,v,w)||||(a,b,c)||\cos(\theta)$ Expanding the 2-norms ...
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1answer
76 views

Find the power series for $d/dx(\arcsin x)$

How would you find the general power series for $\frac{1}{\sqrt{1-x^2}}$ , without using the general rule for arcsinx? I understand it is necessary to use binomial series, but I am having trouble ...
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Taylor series of $\sqrt{1+x}$ using sigma notation

I want help in writing Taylor series of $\sqrt{1+x}$ using sigma notation I got till $1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\ldots$ and so on. But I don't know what will come in ...
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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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3answers
146 views

Taylor Theorem inequality

Prove that for all $f\in C^2([0,1])$ with $f(0)=f(1)=0$ and $|f''(x)| \le 1$ $$|f(x)| \le \frac{1}{2}x(1-x)$$ $\forall x \in [0,1]$.
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2answers
120 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
287 views

taylor series expansion about undefined point

Question is to find coefficient of $(z-\pi)^2$ in taylor series expansion about $\pi$ of $$f(z)=\frac{\sin z}{z-\pi},z\neq\pi$$ $$=-1,z=\pi$$ now, coefficient of $(z-\pi)^2$ should be ...
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Taylor Series on two variables simultaneously

I need to find Taylor series of u($x_i+h,t_n+k)$ where taylor series of $u(x_i,t_n+k)$ and $u(x_i+h,t_n)$ are given as such. Is it basically the summation of both the terms except $u(x_i,t_n)$? ...
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56 views

Application of Taylor's Formula

If we are given that $f''(x) = f(x)$, how do we show that there exist constants $a$ and $b$ such that $f(x) = ae^x + be^{-x}$ for all $x$? A hint is given: We can define another function $g$ by $g(x) ...
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2answers
78 views

Power series of a function about a non zero point

No clue how to ask questions here so here goes nothing! How do I work towards finding the power series of a function centered about a point a not equal to $0$? The specific question I was asked is to ...
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1answer
570 views

Approximation to the square root

I was reading an article that approximated a square root operator as follows $\sqrt{1+x+y} \cong \sqrt{1+x} + \frac{1}{2}y + O(xy,y^2) $ At first glance that looks like a Taylor series expansion, ...
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Inverse of a power series

I want to find the inverse function of the power series, $$ f(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!} $$ The only think I can think of that could possibly help is that $$ ...
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2answers
58 views

Retrieving $f(x)$ from Taylor Series

I've been trying to extract the $f(x)$ from the following Taylor series: $$ \sum_{k=1}^{\infty} (-1)^{k}\frac{kx^{k+1}}{3^{k}} $$ I moved things around a bit to make it look like this: $$ ...
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2answers
45 views

Doubt about Taylor's polynomial to approximate f(x)

Use the degree 2 Taylor polynomial of $f(x) =$ $\sqrt[3]{1728 + x}$ to approximate $\sqrt[3]{1731}$ and give a bound for the error. To obtain the degree 2 Taylor's polynomial, I computed the second ...