# Tagged Questions

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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### Power series as approximation

I have to estimate the error when I approximate the function $$e^{\sin x}$$ to $$1+x+x^{2}+x^{3}$$ when $|x|<0.1$. I really don't know how to do because my teacher didn't teach me. But what I did ...
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### Find the Maclaurin series of f(x)=(arctan(x)-x)/x^3

What I think I need to to do is find a general series expansion of the function and then derive term by term to get the Macaurin series...but I'm not quite sure how to expand this function. Any help ...
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### new equation for $\int_0^ t e^{-x2} dx$? [on hold]

fact! $$\int_0^ x e^{-x^2} dx$$ $$=e^{-x^2}\sum_{n=0}\frac{(2^n)x^{2n+1}}{{(2n+1)!!}}$$
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### Maclaurin polynomial of order 3? Order vs. Degree

I am doing some homework and came across a problem that asks: Find the Maclaurin polynomial of order 3 for f(x) = e^(-4x) When did some searching online, all searches came up as "...maclaurin ...
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### Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
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### (Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
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### Finding the Taylor series of arcsin(1-x)

I'm trying to calculate the Taylor series of $arcsin(1-x)$ about $x=0$. I'm having trouble because I can't compute the derivative there. I can see the correct solution on WolframAlpha (http://www....
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### Taylor Series for a Function of $3$ Variables

The Taylor expansion of the function $f(x,y)$ is: f(x+u,y+v) \approx f(x,y) + u \frac{\partial f (x,y)}{\partial x}+v \frac{\partial f (x,y)}{\partial y} + uv \frac{\partial^2 f (x,y)...
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### Multivariable taylor series approximation

The function is of the form $$F(X) = sum_{i=0}^n x_i*(c_i + ln(x_i/xt))$$ where $X = (x_1,x_2,x_3,...,x_n)$ $xt = sum_{i=0}^n x_i$ $c_i$ is a constant term for ith species I want to find ...
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### Confusion about the different ways of writing Taylor Polynomials

For the sake of using a simple example, let's say I want to approximate $y=x^3$ with a second degree polynomial, and let's say I want to construct my polynomial around the point $x=4$. One way I ...
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### Problem on series expansion and Bessel functions

One way to define Bessel functions is $$e^{\frac{x}{2}(t-\frac{1}{t})}=\sum_{n=-\infty}^{+\infty}J_{n}(x)t^n.$$ How do I prove that? I can't see a way of writing the L.H.S. as a geometrical (or ...
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### Finding the limit of: $\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$ using taylor polynomials

no solution provided so I was hoping someone would do a quick look over and make sure it looks ok. Finding the limit of: $$\lim_{x\rightarrow 0} \left(\frac{1}{f(x)-f(0)}- \frac{1}{xf'(0)} \right)$$ ...
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### Find Polynomial of order 10 for $f(x)=sin(x)$ near x=0

My work so far : I presume the answer should look more like a summation? Thanks!
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### Lagrange remainder vs. Alternating Series Estimation Theorem: do they always give you the same error bound?

Given a function and its nth degree Taylor series approximation, we can use the Lagrange form of the remainder to get a maximum value of the error of approximation. If the series is also an ...
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### Compute $\frac{d^ny}{dx^n}$ if $y = \frac{7}{1-x}$

I am wondering what is $\frac{d^n}{dx^n}$ if $y = \frac{7}{1-x}$ Basically, I understand that this asks for a formula to calculate any derivative of f(x) (correct me if I'm wrong). Is that related to ...
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### Linearization of a function at a point

I have this delay differential equation $$\frac{dx}{dt}=a(x(t)-x(t-1))-b |x(t)|x(t)$$ and I have to make a linearization at the point $\left(\bar{x}(t),\bar{x}(t-1)\right)$, but I cannot figure out ...
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### Singular expansion of an implicit function

In the book of Flajolet and Sedgewick (this context is not so important, though), the following argumentation is used: Let $y(z)$ be a function given implicitly by $y - \phi(z,y) = 0$, where $\phi$ ...
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### How to solve asymptotic expansion: $\sqrt{1-2x+x^2+o(x^3)}$

Determinate the best asymptotic expansion for $x \to 0$ for: $$\sqrt{1-2x+x^2+o(x^3)}$$ How should I procede? In other exercise I never had the $o(x^3)$ in the equation but was the maximum order to ...
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### Implicit Euler using Taylor

I was reading script about differencial equatations. More specific about schemes that help calculate them - implicit Euler. That method was analyzed using something similar to Taylor but i am not sure ...
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### How to find similar convergence rates?

Consider the Taylor's series infinite summation of $\sin(x)$. Let $A_k=\sum\limits_{i=0}^k(-1)^i{x^{2i+1}\over (2i+1)!}$ (Series expansion of $\sin(x)$) I need a series $\{C\}_n$such that its ...
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### How to prove this Taylor expansion of $\frac{1}{(1+x)^2}=-1\times\displaystyle\sum_{n=1}^{\infty}(-1)^nnx^{n-1}$?

I came across this series of the Taylor Expansion- $$\frac{1}{(1+x)^2}=1 - 2x + 3x^2 -4x^3 + \dots.=-1\times\sum_{n=1}^{\infty}(-1)^nnx^{n-1}$$ But I have no idea how to prove this... Thanks ...
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### Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the ...
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### $\Pi_{n=0}^\infty (1-a_n)>0$ if and only if $\sum a_n < \infty$.

Let $a_n$ be sequence in (0,1). $\Pi_{n=0}^\infty (1-a_n)>0$ if and only if $\sum_{n=0}^\infty a_n < \infty$. First I considered $\sum log(1-a_n)$ and tried to find sum inequality. I ...
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### Finding certain coefficients in Taylor expansion of $\log(1 +qx^2 + rx^3)$

This exercise is part of the STEP $3$ paper from $2014$. At a certain point in the problem, we 're supposed find $a_n$ for $n = {2,5,7,9}$ where $a_n$ is the coefficient of $x^n$ in the series ...
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### Elementary Proof of Ramanujan Master Theorem

I was searching for an elementary proof of the Ramanujan Master Theorem and I found a page from Ramanujan's Notebook on wikipedia which contained the proof. I think that it has some gaps, so can ...
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### How to find the asymptotic expansion of $\int_{-\infty}^{y} e^{-x^2/2}/\sqrt{2\pi} dx$ where $x \in N(0,1)$?

I realize the function inside the integral is the pdf of a normally distributed random variable x, but am unsure how to use this to solve the problem. I am trying to relate it to the inverse of the ...
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### When is $1-(1-p)^n \sim pn$

Let $0<p=p(n)<1$ with $p=o(1)$. For which $p$ is it true that $1-(1-p)^n \sim pn$? With $\sim$ I mean that they are asymptotically the same, so $\frac{1-(1-p)^n}{pn}\rightarrow 1$, or at least ...
### Multidimensional taylor series $sin (x^3y^2)$
A homework of mine was to compute the Taylor series of $f(x,y)=\sin(x^3y^2)$ around $(0,0)$ to the 25th order. I assumed, as $\sin(z)=\sum\limits^{\infty}_{k=0}(-1)^k\frac{z^{2k+1}}{(2k+1)!}$, that I ...