1
vote
1answer
14 views

Compostion of functions in taylor series

I was attempting to help answer a question, but I am curious about the following. Suppose we have an analytic function $f(z)\equiv\sum\limits_{k=0}^{\infty} a_k(z-z_0)^k $ on some suitable disk of ...
0
votes
0answers
20 views

Taylor expansion and expansion in powers of z-1

I am trying to expand $z^2/(z+1)^2$ as a Taylor Series. I have acquired its partial fraction decomposition of $z^2/(z+1)^2$ = $(1/6)*(1/(z+1)) + (5/6)(1/(z-5))$. The first term is in the form ...
0
votes
2answers
26 views

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
7
votes
3answers
255 views

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 ...
0
votes
1answer
41 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 ...
3
votes
0answers
72 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 ...
1
vote
2answers
61 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 $2x^2$?
1
vote
1answer
26 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 ...
1
vote
2answers
21 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 ...
0
votes
2answers
46 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: $$ ...
3
votes
2answers
50 views

Finding terms of a Taylor series where $f(x)$ is a function with a power

I've been stuck with this Taylor series problem for a while now. We have that $$ f(x) = (1 + x^2)^{-2/3} $$ and it's centered at $0$. So what I thought of doing was the $$ \frac{f^{n}(a)(x - ...
1
vote
3answers
194 views

Find complicated Taylor Series

According to some software, the power series of the expression, $$\frac{1}{2} \sqrt{-1+\sqrt{1+8 x}}$$ around $x=0$ is $$\sqrt{x}-x^{3/2}+\mathcal{O}(x^{5/2}).$$ When I try to do it I find that I ...
8
votes
1answer
131 views

Taylor Series of $\frac{1}{1-\cos x}$

The problem is, as the title suggests, to find the Power Series Expansion of $\frac{1}{1- \cos x}$ around $x=c$. What I've tried: Direct Computation: Derivatives get very ugly quickly, and don't ...
0
votes
1answer
31 views

Approximation of $\sqrt{1+wi}$

How can $\sqrt{1+wi}$ be approximated? where $-\infty<w<\infty$; My aim here is getting rid of the square root. I've tried binomial, Maclurin and Taylor series around various points. but they ...
0
votes
1answer
35 views

What is the series expansion of $f(z)\cdot\exp\left({s\,\log(z)}\right)$?

For analytic $f$, how can I represent the expression $f(z)\cdot\exp\left({s\,\log(z)}\right)$, i.e. $f(z)\cdot z^s$ in the form $$\sum_{n}^\infty\left(\sum_{k}^\infty a_k s^k\right)z^n,$$ at least ...
3
votes
2answers
79 views

If it converges, how to show that power series converges to $f(x)$?

I had a very basic question. Suppose $f(x)$ is a function. And let us say it has a power series :- $$f(x) = \sum_{n=0}^\infty a_nx^n.$$ Suppose we are operating inside the region of convergence. ...
0
votes
1answer
36 views

A power approximation function

I am trying to construct a function that would approximate $a^b$ using Maclaurin series. Here are my reasoning: Since $$a^b=e^{b\ln a}$$ and $$e^x=\sum^{\infty}_{k=0} \frac{x^k}{k!}$$ it should ...
0
votes
1answer
34 views

Help with Taylor series problem

I am using maple to plot the graphs of e^e^x versus its truncated Taylor series around 0. For small values of x, the two graphs converge nicely, but once x<-3, my Taylor series loses control. Here ...
1
vote
1answer
94 views

Taylor series of $\frac 1 {1+x^2}$

I have to construct the Taylor series of $$\frac 1 {1+x^2}$$ around $0$ and $1$ and analyze the convergence in both cases. Also (but this is a consequence of the previous series) I have to construct ...
0
votes
3answers
49 views

A simple series

I don't do math a long time, so I completely don't remember how to prove that: $$ \sum_{i=1}^\infty \frac{i}{2^i} = 2 $$ Can anybody help me?
4
votes
2answers
158 views

Express $(1-z)^{-1}$ as a power series around $z_0=-1+i$.

I need to express $(1-z)^{-1}$ as a power series in powers of $(z+1-i)$. I would like some guidance on the complex analogue of power series and in writing out this particular case. Many thanks for ...
3
votes
1answer
64 views

Showing $|a_k | \le 1$

Let $A$ be the closed unit disk $A= \{z \in \mathbb{C}: |z| \le 1 \}$. Suppose $f$ is an entire function whose Taylor series centered at the origin is $$\sum_{k=0}^{\infty} a_kz^k$$ and that $f$ maps ...
0
votes
2answers
30 views

Find the Maclaurin series of $f(x)=\frac{x}{x^4+x^2+1}$ [closed]

Maclaurin series for $f(x)$. Thanks.
1
vote
1answer
26 views

Expanding One Function in Powers of Another

One sees here that it is possible to expand $f(x) = 2x^3 + 7x^2 + x - 6$ in powers of $x - 2$ by taylor expanding $f(x) = f(x - 2 + 2) = f(2 + h)$ about $2$, and this idea can be used in deriving the ...
0
votes
2answers
43 views

how can I sove approximation evaluation of this integral?

$$\int_{-1}^{0}\sin(e^{x})\,dx $$ approximation of this formula up to difference(error) $1/5000$ Because of the error size $1/5000$ , I think it's solved by taylor expansion.
2
votes
1answer
99 views

Convergence radius of $\sqrt{\cos(z)}$

Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root. I've computed ...
1
vote
1answer
54 views

Maclaurin Series of $\int_0^x \cos t^2\,dt$

Find the Maclaurin Series for $\int_{0}^{x}\cos t^2\,dt$. $$\cos(x) = \sum\frac{(-1)^n x^{2n}}{2n!}$$ I'm trying this: $$\cos^2 x = \sum\frac{(-1)^n x^{4n}}{(2n!)^2}$$ How would you solve this ...
1
vote
3answers
170 views

First $3$ non-zero terms of the Maclaurin Series $\frac{1}{\sqrt{4+x^3}}$

Since each derivative will be multiplied by $3x^2$, are all the terms of this Maclaurin series $0$?
0
votes
0answers
39 views

Inequality sine power series (induction)

How can we show, for $k\geq 1$ and $x \geq 0$, the inequality below by induction? $\displaystyle \sin x \geq \sum_{n=1}^{2k} (-1)^{n+1} \frac 1{ (2n - 1)! }x^{2n-1} $ The base case $k = 1$ gives ...
1
vote
0answers
161 views

Taylor's Formula vs. Taylor's Inequality

In my calculus book, Essential Calculus, and in class we were using Taylor's formula to approximate the remainder in Taylor polynomials but I am having a bit of trouble understanding the intuition ...
1
vote
0answers
41 views

Taylor's Formula and 'z' values

I have attached Taylor's Formula, an exercise problem from a section on Taylor polynomials, and the solution to this exercise. I understand part a, expanding $f$ using Taylor polynomials is the ...
1
vote
1answer
82 views

How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...
1
vote
0answers
50 views

Radius of convergence for Taylor series?!

Given is: $f(x) = \frac{\sin x}{x} $ I need the Taylor series in $a = 0$, so: $$T(x,0) = \frac{1}{x} \sum_{n=0}^\infty ((-1)^n* \frac{x^{2n+1}}{(2n+1)!} ) = \sum_{n=0}^\infty (-1)^n * ...
-2
votes
1answer
27 views

Coefficients of Maclaurin Series

Here's the prompt: http://imgur.com/iAqVWI6 I can find the Maclaurin series, but I'm having trouble finding Cn. Can anyone help please?
4
votes
1answer
125 views

Maclaurin series for $e^z /\cos z$.

I want to find the Maclaurin series for the function $$f(z)=\frac{e^z}{\cos z}.$$ Right away I can tell that the radius of convergence will be $\pi/2$, since it's the distance to the nearest ...
2
votes
1answer
129 views

Complex Analysis - Taylor Series

The question reads as follows: Let the function $f$ be given by $f(z)=\frac{z}{2}+\frac{z}{e^z-1}$ if $z\neq0$ and $f(z)=1$ if $z=0$. Show that $f$ is analytic at $z=0$ and that $f(z)=f(-z)$. ...
2
votes
1answer
164 views

Intuition regarding Taylor series for $\frac{e^z}{1-3z}$.

The question asks me to find the Taylor series for $$f(z)=\frac{e^z}{1-3z}.$$ The radius of convergence is $|z|<1/3$ and I know the expansions for $e^z$ and $1/(1-3z)$ are \begin{align} e^z ...
4
votes
2answers
174 views

Taylor series Question

So I have a test next week and I saw this question with no answer and I would like to some help. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ infinitely differentiable and let $\sum _{n=0}^{\infty} ...
4
votes
1answer
89 views

Calculating $\arctan(3)$ using Taylor series

I'm trying to get a Taylor series equivalent for $\arctan(3)$, but the standard definition for $\arctan(x)$ is restricted to $|x| \le 1$. How can I get a Taylor series for this expression?
0
votes
1answer
98 views

Series Expansion of an Exponential with a Trig Function in the Exponent

Can anyone get a general expression for $$e^{a\cos x}$$ in terms of an infinite sum? I'm having trouble with a general form in terms of $n$ for the coefficients... Alex
1
vote
0answers
81 views

$\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}x^k =$

Prove that for $|x|<1$, $\sum_{k=0}^{\infty}\frac1{4^k(2k+1)}\binom{2k}{k}x^{2k+1}\sum_{k=0}^{\infty}(-1)^k\frac1{4^k}\binom{2k}{k}(-x^2)^k = ...
0
votes
1answer
75 views

What's Taylor expansion of: $f(x)=\frac 1x\ln{(1+2x^2)}$?

What's Taylor development on the next function: $f(x)=\frac 1x\ln{(1+2x^2)}$? Actually this one is the first question I've seen with $ln$, My instincts tell me to try and do derivative in order to ...
1
vote
1answer
79 views

Taylor series of a power series.

Consider a power series $f(x)$ around a point $c \neq 0$. Then is it equal to its Taylor series around $0$? The reason I am wondering about this is because if it is true even for some special cases, ...
2
votes
2answers
252 views

Induction Proof for a series expansion of a function

I have done induction proofs of many different types, but trying to prove by induction that a derivative from the Taylor series expansion of a function has me stumped in terms of how to get the final ...
0
votes
1answer
97 views

How do we know that $\sum_{k=0}^{\infty}\frac{x^k}{k!}=e^x$?

I've been taught that the definition of the exponential function is the following power series: $$\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ Here's my question: how do we know that this series is equal to ...
1
vote
1answer
99 views

Defining square root by series, and showing properties.

I'm triyng to define the root of a complex number near $1$ using the Taylor's series for $\sqrt{1+x}$, but I'm having some problems. Let $x\in\mathbb{C}$ such that $|x|<1$. Let ...
1
vote
1answer
41 views

Rearranging power series expansion to get parameter on denominator

How can we rearrange $$T=\dfrac{k V+g}{gk}\bigg(kT-\dfrac{1}{2}k^{2}T^{2}+\dfrac{1}{6}k^{3}T^{3}\bigg),$$ to get $$T=\dfrac{2V/g}{1+k V/g}+\dfrac{1}{3}k T^{2}$$ ?
62
votes
3answers
1k views

Is the derivative the natural logarithm of the left-shift?

(Disclaimer: I'm a high school student, and my knowledge of mathematics extends only to some elementary high school calculus. I don't know if what I'm about to do is valid mathematics.) I noticed ...
11
votes
2answers
479 views

Taylor series (or equivalent at $\epsilon\to0$) of the integral over $x$ of a function of $x$ and $\epsilon$

I have a function $f$ of two arguments, defined as $$ f(x,\epsilon)=\epsilon\left( e^{-\frac{(x-\epsilon)^2}{2}} - e^{-\frac{x^2}{2}}\right) + \frac{1-\epsilon}{\sqrt{1+\epsilon}}\left( ...
2
votes
1answer
67 views

Find Taylor series expansion and convergence radius for $\int_0^x\cos(\sqrt{t}\ )dt$

i must find the the Taylor series expansion (i've been asked not necessarily calculating it directly) and the convergence radios for this function : $$f(x) = \int_0^x \cos(\sqrt{t}\ ) \, dt$$ I am ...