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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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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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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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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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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How are the Taylor Series derived?

I know the Taylor Series are infinite sums that represent some functions like $\sin(x)$. But it has always made me wonder how they were derived? How is something like ...
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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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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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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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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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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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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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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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604 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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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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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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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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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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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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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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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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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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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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28 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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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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Simplest proof of Taylor's theorem

I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem. By which I mean this: there are plenty of proofs that introduce some arbitrary construct: ...
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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
602 views

Computing the elements of a Hessian matrix with finite difference

I have a generic function $g(x)$ where $x$ is an 6-dimensional vector. Now I want to compute the Hessian of this function for a point $x_0$. What is the most efficient way to do this? Can I do this ...
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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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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
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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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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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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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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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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
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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Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?

Using the Taylor expansion $$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain $$f(x+a) = ...
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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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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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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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Taylor Series of Integral

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