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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66 views

Solve limit of integral through taylor

Show using Taylor expansion that $$\lim_{r\to 0} \frac4{\pi r^2} \int_0^{2\pi} f(a+r\cos \theta , b +r\sin \theta)\cos{2\theta}d\theta = f_{xx} (a,b) - f_{yy}(a,b)$$ where $f:\mathbb R^2 \to ...
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1answer
69 views

Find the Taylor Series expansion of the given analytic function

Find the Taylor Series expansion of the given analytic function $f(z)$, centered at point $z_0$; find the disk of convergence. a) $f(z)=\frac{1}{-2+3i-z}$ $z_0=3$ b) $f(z)=(2-z)\cos{(3z^2)}$ ...
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2answers
41 views

taylor series for two variables

The theorem I have been given for this is $$f(x,y)=f(a+u,b+v)=f(a,b)+\sum \limits_{k=1}^{\infty} \frac1{k!} \bigg(u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}\bigg)^kf(a,b)$$ where ...
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1answer
71 views

Find the Taylor series expansion (centered at $z_{0}=0$) of the function $f(z)=\sin(z^3)$.

Find the Taylor series expansion (centered at $z_{0}=0$) of the function $f(z)=\sin(z^3)$. Use this expansion: a) to find $f^{69}(0)$; b) to compute the integral transversed once in the positive ...
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2answers
38 views

Expand the function in a Maclaurin series $\ln(5\cos^{3}(x))$

$$\ln(5\cos^{3}(x))$$ Need to be expanded: $x^{4}$ I need to end this problem. So I laid the beginning of the function. $$\cos x=1-\frac{x^2}{2!}-\frac{x^4}{4!}+o(x^4)$$ ...
2
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1answer
59 views

Reverse engineering a Taylor expansion 2

So there is the sum: $$S(x) = \frac{x^3}{3(1!)} + \frac{x^6}{6(2!)} + \frac{x^9}{9(3!)} \text{ }...$$ and we are instructed to find the sum of the series in a small expression. I took the derivative ...
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0answers
68 views

Showing a hyperbola in polar form approaches two asymptotes

Consider a curve given in polar coordinates by $r(θ)=\frac{1}{1+e\cos\theta}$, where $e≥0$. When $e>1$, show that the curve approaches two asymptotes, find them and sketch the curve. Hint: If the ...
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0answers
49 views

Taylors formula using little-o notation proof argument (continuity)

Im trying to prove the following: Let $f: I \to \mathbb{R}$ be $C^n$ on $I \subset \mathbb{R}$ and $P_n$ be the $n$'th degree Taylor polynomial with $a$ as the expansion point then $$ f(x) = P_n(x) ...
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28 views

Taylor series of Lagrangian

Take a look at the Lagrangian defined here. $L=\frac12 a(q)\dot q^2 - V(q)$. You can think of $a$ and $V$ as functions. It seems as though $L$ depends only on $q$. If $q_0$ is a point for which ...
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0answers
45 views

Finding a Taylor Expansion for the following:

I have: $$\frac{1}{1-z}$$ for $z_0=i$. I have no idea how to do the Taylor Series expansion of this, around $z_0=i$, and then show it summation form. I have: $\frac{1}{1-z} = ...
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1answer
64 views

Find the Taylor expansion of $\frac{1}{x^2+2x-3}$ around $x=-1$.

Find the Taylor expansion of $\frac{1}{x^2+2x-3}$ around $x=-1$. What is its radius of convergence? So I write the fraction as $\frac{1}{(x-1)(x+3)}$ and what should I do now?
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1answer
24 views

Taylor Expansion of Inverse of Difference of Vectors

I am trying to derive the multipole moment of a gravitational potential, but I'm getting stuck on some math I believe. So basically the problem is finding the Taylor Expansion for ...
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1answer
88 views

Taylor series of mandelbrot bulb boundaries

What I am looking for is a way to find an approximation to the boundaries of hyperbolic components of the Mandelbrot set. I would like to be able to write a program to find the equations which ...
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0answers
258 views

Differences among Cauchy, Lagrange, and Schlömilch remainder in Taylor's formula: why is generalization useful?

I would like to know what really are the main differences (in terms of "usefulness") among Cauchy, Lagrange, and Schlömilch's forms of the remainder in Taylor's formula. Could you provide examples ...
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2answers
63 views

Taylor series Expansion

I'm a little confused as to what they are asking. all the examples of taylor series expansion I have seen use x instead and I'm not sure how I would go expanding these series.
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26 views

Taylor series of $(1-x)^b$ $_2F_1(a,b;c;x)$: when to stop?

Let $f(x)= (1-x)^b$ $_2F_1(a,b;c;x)$, where $0<x<1$ and $a=(K-1)d$, $b=K$, c=$Kd$ (with $a$, $b$ and $c$ are positive and $K>d$ ). I need to derive the Taylor series of the corresponding ...
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1answer
62 views

The Taylor coefficients of a function of the form $\exp\circ f$, where $f$ is a power series

Let $(a_1, a_2, \dots) \in \mathbb{R}^\infty$ be a fixed sequence of real constants, and suppose the rule $$ x \mapsto \sum_{n = 1}^\infty a_n x^n $$ defines a function from the nonempty open interval ...
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0answers
32 views

General form for series coefficient of Taylor series expansion of $(x+1)^{1/x}$

What is the general form for the series coefficients of Taylor expansion of $(x+1)^{1/x}$? The first few terms are as follows: $$e-\frac{e x}{2}+\frac{11 e x^2}{24}-\frac{7 e x^3}{16}+\frac{2447 e ...
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3answers
61 views

Taylor Series for $\frac{1}{1+e^z}$ and radius of convergence

I have done some manipulation and got that $$\frac{1}{1+e^z} = \sum_{n=0}^\infty \frac{n!}{n!+z^n}$$ by the fact that: $$\frac{1}{1+e^z}= ...
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2answers
46 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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14 views

Effective ways to calculate multivariable taylor expansion

I need to calculate first 20 members of taylor series for $e^{x^7+y^{11} \cos{(x^{10}+y^8})}$. Are there any ways except the terrible direct way.
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269 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 ...
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2answers
510 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
130 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 ...
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1answer
22 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 ...
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3answers
35 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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1answer
50 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
36 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
54 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 ...
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0answers
44 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
36 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) ...
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1answer
46 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 ...
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4answers
151 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 ...
2
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1answer
72 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 $C(1+i)^n$ ...
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4answers
77 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
51 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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39 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 ...
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1answer
44 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 ...
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1answer
42 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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3answers
68 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
966 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
136 views

Evaluate $I=\int_{\gamma (0,1)}\frac{zdz}{(z^2-4z+1)^2}$ using Taylor's Theorem

I've shown that $f(z)=\frac{z}{(z^2-4z+1)^2}$ is holomorphic apart from at points $\alpha=2-\sqrt3$ and $\beta=2+\sqrt3$ and that the talyor coefficient of $g(z)=\frac{z}{(z-\beta)^2}$ centred at ...
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2answers
37 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 ...
3
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2answers
59 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 ...
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1answer
32 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 + ...
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1answer
53 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 ...
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2answers
53 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 ...
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1answer
41 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 ...
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0answers
92 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 $n$" , ...
3
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1answer
73 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 ...