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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41 views

Differentiate a Differential equation

Given the Differential equation $y'=-2xy^{2}$. Find the derivative $\frac{d(y')}{dx}$! My approach, which is not correct according to Wolfram Alpha: Plugging in: ...
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0answers
36 views

Trigonmetric calculus, [duplicate]

Why is the macluaren representation for cos and sine in radians and not degrees, isnt the deravative on cos(x) and Sin(x) in both degrees and radians equaly -sin(x) and cos(x)?
3
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2answers
464 views

Exponential function-like Taylor series: what is it?

I have a series $$1+ x+\frac{x^2}{2}+\frac{x^3}{4}+\frac{x^4}{8}...=1+\sum_{n=1}^\infty \frac{x^n}{2^{n-1}}$$ that looks an awful lot like a Taylor series of some kind. If the denominator of the ...
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1answer
16 views

Taylor expanion of exponential matrix

I've been reading about Lie groups, and came across the following expansion that left me confused: Let $$ A = e^{i\lambda X_a} \text{ and } B = e^{i\lambda X_b} $$ for matrices $X_a$ and $X_b$, and ...
2
votes
3answers
30 views

Errors and Taylor Polynomials

For $g(x)=x^{1/3}$, $a=1$, degree $3$ I found the Taylor polynomial: $$p_3(x) = 1 + (x-1)/3 - ((x-1)^2)/9 + (5(x-1)^3)/81$$ How do I use the error formula for the Taylor polynomial of degree 3 to ...
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vote
1answer
60 views

Ring of Infinitely Differentiable Functions

Denote the ring $\text{C}^{\infty}$ i.e. infinitely differentiable functions from $\mathbb{R}\mapsto\mathbb{R}$. I have managed to prove that this ring is a Principal Ideal Domain. I must also prove ...
-1
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1answer
41 views

Show by comparison with an appropriate geometric series that, $e^x-1<\left(\frac{2x}{2-x}\right)$ for 0<x<2.

Show by comparison with an appropriate geometric series that, $e^x-1<\left(\dfrac{2x}{2-x}\right)$ for $0\lt x\lt2$. Can anyone help me with this?
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1answer
19 views

Evaluating irrational values of functions with Taylor series

Calculate the following using Taylor expansion such that the error will be smaller than $10^{-3}$. $\tan 46^\circ$ $(31)^{1/5}$ My problem is that I don't know if I can avoid to use ...
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1answer
44 views

Approximating $e^{\frac 1 {10}}$ with Taylor expansion

Approximate $e^{\frac 1 {10}}$ such that the error won't be larger than $10^{-3}$. I tried to use the expansion for $e^x$ but the error is too large even beyond order 4. So I think the only ...
0
votes
0answers
37 views

Estimating the error of $\sin x \approx x-\frac {x^3}{6}$ for $|x|\le \frac 12$

Estimate the error of $\sin x \approx x-\frac {x^3}{6}$ for $|x|\le \frac 12$ It seems too easy so I just want to make sure: Since $f(x)-p(x)\le R(x)$ and $R_5(x)=\cos (c) \frac {x^5} {5!}$ So ...
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1answer
34 views

Compute the 100th Bernstein polynomial for $e^x$

I need to find $$B_3 e^x = \sum_{k=0}^{100} e^{k/100}\binom{100}{k} x^k (1-x)^{100-k}$$ I can rearrange this to find $$\sum_{k=0}^\infty e^{k/100} \left(\frac{100!}{k!(100-k)!}\right) ...
0
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1answer
42 views

Taylor Series Clarification

For $\sin(x)$, $e^x$, $\cos(x)$... When we are building the $n$-th taylor polynomial, why is it that we always evaluate the functions first $k$ derivatives at $x=0$? In my textbook when they were ...
5
votes
1answer
122 views

What is the connection between Taylor series and Chebyshev polynomials?

Can somebody help me find some historical references for the connection between Chebyshev polynomials and the Taylor series for sine and cosine functions? We know that Chebyshev polynomials are used ...
2
votes
1answer
35 views

Simple vs compound interest rates and Taylor expansion

I am having trouble deciphering a portion from my finance text. Let $i = \text{interest rate}$, $n = \text{Some arbitrary time period}$ and $C = \text{Cash invested}$ And also $C(1+i)^n$ ...
2
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4answers
73 views

Find a power series for this function

$$f'(x) = 2xf(x) + 4x$$ I need to find the power series for $f(x)$, any hints on how this should be approached?
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0answers
42 views

Proving substitution rule of taylor series

Given $f, g$ which are both nth differential-able. How do I show that $f(g)$ is also nth differentai-able ? I tried using chain rule to calculate, but it seems like a mess. Then how can I show that ...
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0answers
29 views

why does this power series converges to sinh(x)?

given the infinite sum $$\sum_{n=0}^\infty \frac{ x^{2n+1}}{(2n+1)!}$$ of course, by ratio test, it converges for reals. I know that the answer is $\sinh(x)$ and I've seen how this is derived from its ...
0
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1answer
30 views

Prove that $\cos(x^2)$ is analytic at $x = 0$

I can't figure out how to go about showing any of the properties required for analytic with such a messy derivative. This is for my real analysis class and I just want to see for this example so that ...
9
votes
4answers
134 views

Bernoulli Number analog using Cosine

I know that Bernoulli Numbers can be found with the generating function $$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$ I was wondering if any work has been done using a similar equation ...
0
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3answers
65 views

Finding $\sum\frac{1}{2^n(n+1)}$

What is the sum of $$\sum_{n=0}^{\infty}\frac{1}{2^n(n+1)}$$ I've spent an insane amount of time on this problem. I checked on Wolfram and it gives $ln(4)$, which I assume you get from $2\ln(2)$. ...
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4answers
514 views

Taylor series of ln(1/(1-z)) around 0

One more taylor/maclurian series problem to which I know the answer of, I just have no idea how to get there (This is as a formal power series, so convergence is not an issue) $$\log \left(\frac 1 ...
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1answer
30 views

N'th coefficient of two taylor series

So, I'm taking a course in Analytic Combinatorics, and the author asserts without proof that the n'th coefficient of $z^n$ for the taylor (Around 0) expansion, for nonnegative integer values of r in ...
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2answers
28 views

Expand a function in Maclaurin's series.

The function is given with: $$\ln(5\cos^{3}(x))$$ Need to be expanded: $$x^{4}$$ I have no idea what to do here and honestly, I don't really understand what a Maclaurin's series is. I know the ...
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1answer
27 views

Taylor Series for $e^x(x^2 -x + 1)$

Find the Taylor Series for $e^x(x^2 -x + 1)$ about $x=0$. More importantly, find the COEFFICIENT (for nonzero terms) of the taylor series. The answer says: $$e^x(x^2 -x + 1) = 1 + ...
3
votes
2answers
51 views

function approximation using series

I was trying to approximate a function by another one using some kind of a series. Let $$ f(x) = m\cdot \left(1-\sqrt{\frac x2\biggm/\left(1+\frac x2\right)}\right) $$ I'm trying to approximate ...
0
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1answer
19 views

For small x, Taylor Series to determine constants n and C in the approximation

cos(x) - e^(-(x^2)/2) I have tried writing the taylor expansion for cosine and e but do not know how to combine them in one. In addition to this, i do not know how to approach the problem whilst ...
0
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1answer
35 views

Why can you use the Maclaurin Series for certain cases of function not about 0?

Is it possible to use the Maclaurin Series in a problem like this one (AP Calculus BC Question 6 from a few years ago)? Write the first four nonzero terms and the general term of the Taylor ...
2
votes
2answers
52 views

Taylor series of $x/(x^2-4x+5)$

I'm supposed to find the Taylor series of this function (I can choose to center it at any A I want): $$f(x)= x/(x^2-4x+5)$$ When I derivate, it only gets more and more confusing. How can I make any ...
2
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1answer
50 views

Convenient notation, or something more?

A little while ago I happened across a curious formula that blew my mind (no idea what it's called): $e^{\frac{d}{dx}}f(x)=f(x+1)$ I played around with it a bit and managed to prove it using the ...
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1answer
31 views

Taylor Series to the Power 1/z

I am attempting to find the Taylor Series for $(\frac{\sin{z}}{z})^{\frac{1}{z^2}}$. While I can plug this into Wolfram and use the output, I want to understand how to calculate the Taylor Series ...
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1answer
27 views

Is there an interpretation for writing a polynomial in $x$ as a polynomial in $(x-b)$?

Let $Q(x)$ be a polynomial in $x$ of order $n$. The Taylor polynomial of $Q(x)$ of order $n$ developed around $x=b$ (denoted by $P_{n,b}(x)$ ) corresponds to $Q(x)$ written in $(x-b)$. This can be ...
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1answer
37 views

Messing with the linear approximation… I don't get Taylor's formula??

Alright, so I was messing around with the linear approximation for a function; I wanted to see if I could get the formula for the 2nd degree approximation from it. I went about it like this: $$ f(x + ...
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0answers
17 views

A formal solution using Taylor series

Consider the following: $S_r(n)= 1^r+2^r+...+(n-1)^r$ where $S_r(n)$ satisfies: $S_r(n+1)-S_r(n+1)=n^r$ Now, also consider the Taylor series $f(x+h)=f(x)+hf'(x)+(h^2/2!)f'(x)+..$ which can be ...
0
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2answers
22 views

Taylor Series with differentiator operator

Hi guys can anyone show me how the Taylor series can be converted from: $$f(x+h)= f(x)+hf'(x)+...$$ to: $f(x+h)=e^{hD}f(x)$, where $D$ is the differentiation operator. How does the differentiator ...
0
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2answers
19 views

expand function, taylors series, combinatorics, generation functions

I have to expand $f(z)$ into a formal power series $f(z) = \sum\limits_{k=0}^\infty a_kz^k$ (for $z$ close to 0) $f(z)= \frac{z^3}{1-4z+3z^2}$ I know that: $\frac{1}{1-z} = \sum\limits_{k=0}^\infty ...
0
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1answer
14 views

Equality form of second order Taylor series

I am reading a book on optimization wherein a statement using Taylor's expansion is made without proof. \begin{equation} f(\mathbf{y}) = f(\mathbf{x}) + (\mathbf{y} - \mathbf{x})^T\nabla ...
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1answer
36 views

Need clarification on a Taylor polynomial question

$$f(x) = 5 \ln(x)-x$$ second Taylor polynomial centered around $b=1$ is $-1 + 4(x-1) - (5/2)(x-1)^2$ let $a$ be a real number : $0 < a < 1$ let $J$ be closed interval $[1-a, 1+a]$ find upper ...
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1answer
243 views

Erroneously Finding the Lagrange Error Bound

Consider $f(x) = \sin(5x + \pi/4)$ and let $P(x)$ be the third-degree Taylor polynomial for $f$ about $0$. I am asked to find the Lagrange error bound to show that $|(f(1/10) - P(1/10))| < 1/100$. ...
2
votes
1answer
53 views

Complex Taylor Series Circles of Convergence

I am trying to find the Taylor Series and circles of convergence for three different functions. i) $\frac{\sin{z}}{z}$ which I determined the Taylor series to be $\sum_{n=0}^\infty ...
0
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1answer
32 views

What does the h mean within the Taylor expansion of $y(x_0 +h)$ and $y(x_0 -h)$?

I understand that the Taylor series formula is $$\frac{f^n(a)}{n!}(x-a)^n.$$ I also know that the Taylor series expansion of $$y(x_0 +h)=y(x_0) +hy'(x_0)+\frac{h^2}{2!}y''(x_0)+ ...
0
votes
1answer
52 views

first order approximation of scalar function of matrix ( Mahalanobis distance)

I have tried to compute the 1st order approximation using Taylor's expansion of the Mahalanobis distance: $f(\mathbf{X})=\mathbf{a^TXa}$, where $\mathbf{a}\in \mathbb{R}^N$. The function maps ...
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1answer
29 views

Find Taylor Polynomial of degree $2$ about (2,1) where $f(x,y) = x^2y^3$ , $(x,y) \in \mathbb{R}$

Find Taylor Polynomial of degree $2$ about $(2,1)$ where $f(x,y) = x^2y^3$ $(x,y) \in \mathbb{R}$ My thoughts: $D_xf= 2xy^3$ and $D_{xx}f= 2y^3$ $D_yf= x^23y^2$ and $D_{yy}f= x^26y$ $D_{xy} = ...
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1answer
73 views

Gibbs Phenomenon and Fourier Series

a) Show the partial sum $$S = \frac{4}{\pi} \sum_{n=1}^N \frac{\sin((2n-1)t)}{2n-1}$$ which may also be written as $$ \frac{2}{\pi}\int_0^x\frac{\sin(2Nt)}{\sin(t)}dt$$ has extrema at $x= ...
0
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0answers
65 views

Beyond taylor series?

Consider functions $f(z)$ that are analytic for $Re(z) > 0$ and are also analytic for $(Im(z))^2 > 0$. Let $n$ be a nonnegative integer. Now I define some series expansion of "order $n$" , ...
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1answer
57 views

Using Taylor's Theorem and the Constancy Theorem, solve the following proof.

Using Taylor's Theorem and the Constancy Theorem prove that $\sqrt{1+x}=1+\frac{x}{2}+\sum_{n=2}^{\infty}(-1)^{n-1} \frac{1}{2n} \frac{(1- \frac{1}{2})(2- \frac{1}{2}) ... ((n-1)- ...
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0answers
22 views

Function approximation by various means

I know several ways to approximate a function: Taylor series, Fourier series, or polynomials, like e.g. Legendre polynomials. Is the only difference between those various methods the speed at which ...
2
votes
4answers
52 views

Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$.

Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$. Taylor's Theorem applies at the point $a=0$ and with $n=4$. Got no idea how to proceed. My lecture notes have one example ...
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2answers
43 views

Trying to determine the Inverse function of $\sinh$ and $\cosh$

I'm trying to find out how to determine the inverse function in order to develop the $$ \sinh(x).$$ I tried to expand to its exponential form $$\sinh(x) = \frac{1}{2} (e^x-e^{-x}) .$$ So I wrote $$ ...
1
vote
1answer
26 views

Is it Possible to Develop an inverse function using the function it self

Is it Possible to Develop (taylor expansion) of an inverse function by knowing the function it self ? If Yes ,Can you illustrate with a simple function I know that we use the identity formula $$ ...
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2answers
53 views

Estimate ln(3) using Taylor Expansion up to 3rd order

Estimate ln(3) using Taylor Expansion up to 3rd order (without the use of a calculator). $$f(x)=ln(x)$$ $$f'(x)=1/x$$ $$f''(x)=-1/x^2$$ $$f'''(x)=2/x^3$$ ...