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.

learn more… | top users | synonyms (1)

4
votes
0answers
195 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 ...
4
votes
0answers
60 views

Manipulation of Taylor/Laurent series

I have a question regarding how to expand a given rational function into its Taylor/Laurent series representation. Suppose we are given the function $$f(z) = \frac{z}{(z-1)(z-3)},$$ and are asked to ...
4
votes
0answers
248 views

Taylor Formula: Lagrange's remainder vs Cauchy's remainder (and other less known forms)

While solving problems and exercises, so far I've only used Lagrange's form of the remainder. Indeed, it must be said that many textbooks don't even mention other forms of the remainder for Taylor's ...
4
votes
0answers
246 views

A late-diverging “approximating solution” for a system of functional equations

Peace be upon you, At the end of this question, I have shown that how computing MLE on an i.i.d Beta distributed data, results in the following system \begin{align*} &\begin{cases} ...
4
votes
2answers
88 views

Approximating the erf function

I was trying to find an approximate solution to the following: $\DeclareMathOperator\erf{erf}$ $$\frac12 \sqrt{\pi} \erf\left (\frac{x-2}{\sqrt{10}}\right) + \frac12 \sqrt{\pi} \erf ...
4
votes
0answers
219 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 ...
4
votes
0answers
90 views

Inverse of a power series

I want to find the inverse function of the power series, $$ f(x)=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n+1)!} $$ The only think I can think of that could possibly help is that $$ ...
4
votes
1answer
90 views

How to find the series $\sum_{n=1}^{\infty}\frac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$

Find this sum $$\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n}.\qquad (-1\le x\le 1)$$ My idea: let $$f(x)=\sum_{n=1}^{\infty}\dfrac{[(n-1)!]^2}{(2n)!}(2x)^{2n},$$ then we have ...
4
votes
0answers
91 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
4
votes
0answers
48 views

Am I missing anything when doing this taylor expansion?

I'll make my question short, I am encountering a error when doing expansion. I am expanding $f(x)=2x^3+4x+1$ and after the expansion, things don't match. Here's what I'm doing. Let $a=5$ ...
4
votes
2answers
430 views

Range of the sine function

It is obvious from the definition of $f(x)=\sin(x)$ using the unit circle of radius $1$ that the range of that function is the set $[-1,1]$. But also there are approaches where the sine is defined ...
4
votes
1answer
162 views

Write down the equation of the tangent plane and compute the Taylor series of the function

Set $f(x,y,z) = x + y + z + x^2 + y^2 + z^2$. Consider the surface $$S = \{f(x,y,z) = 0\} \subset \mathbb{R}^3$$ near the origin $o = (0,0,0) \in S$. Write down the equation of the ...
4
votes
1answer
118 views

A sort of “Taylor expansion” of a power series

I have the following question. Suppose $$f(x):=\sum_{i=0}^{\infty}c_ix^i$$ is a power series that converges for $|x|<1 + \epsilon$, for some $\epsilon >0$, where $x\in\mathbb{C}$. I can then ...
4
votes
0answers
460 views

Fastest convergence Series which approximates function

The question is the following: Is there any proof that shows that the Taylor series of an analytical function is the series with the fastest convergence to that function? The motivation to this ...
3
votes
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 ...
3
votes
5answers
121 views

Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that ...
3
votes
3answers
626 views

Taylor series for different points… how do they look?

I can't understand what it means to do the Taylor series at the point $a$. The best way would be showing me how it looks for different $a$ on a graph. Do I find those graphs on the Internet?
3
votes
4answers
4k views

Maclaurin Series of $1/(1-x)$ derived from maclaurin series of $(1+x)^n$

Is there a way to derive the Maclaurin series for $\frac{1}{(1-x)}$ after finding the Maclaurin series for $(1+x)^n$ which is $\displaystyle\sum\limits_{k=0}^\infty \frac{f^k(0)}{k!}*x^k$. From ...
3
votes
5answers
102 views

When $a\to \infty$, $\sqrt{a^2+4}$ behaves as $a+\frac{2}{a}$?

What does it mean that $\,\,f(a)=\sqrt{a^2+4}\,\,$ behaves as $\,a+\dfrac{2}{a},\,$ as $a\to \infty$? How can this be justified? Thanks.
3
votes
2answers
233 views

Taylor Series of $\tan x$

I found a nice general formula for the Taylor series of $\tan x$: $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$ where $B_n$ are the Bernoulli ...
3
votes
2answers
102 views

Show that $f(x)=e^x$

In this case $f(x)=1+x+x^2/2!+x^3/3!+x^4/4! + ... = \sum_{n=0}^\infty \frac{x^n}{n!}$. I understand it conceptually in terms of the Taylor series, but I have no idea how to prove it rigorously.
3
votes
2answers
59 views

Finding the sum of a Taylor expansion

I want to find the following sum: $$ \sum\limits_{k=0}^\infty (-1)^k \frac{(\ln{4})^k}{k!} $$ I decided to substitute $x = \ln{4}$: $$ \sum\limits_{k=0}^\infty (-1)^k \frac{x^k}{k!} $$ The first ...
3
votes
2answers
582 views

Why is Taylor series expansion for $1/(1-x)$ valid only for $x \in (-1, 1)$?

After finding an expansion of $$ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots$$ a quick test of various values for $x$ reveals that this expansion is not valid for $\forall x \in \mathbb{R}-\{1\}$. ...
3
votes
2answers
458 views

Odd powers in the Taylor series expansion of $\operatorname{Log}(1+e^z)$.

I noticed a question while reviewing Taylor series expansions that has been bugging me. The question is: How many nonzero terms with odd exponents does the Taylor series $\operatorname{Log} (1 + e^z)$ ...
3
votes
3answers
118 views

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… ...
3
votes
2answers
106 views

Question involving approximation, taylor series and proving

Question: Consider the approximation $$\ln(2)\approx 2\left ( \frac{1}{3}+\frac{1}{3\times 3^{3}}+\frac{1}{5\times 3^{5}} \right )$$ Prove that the error in this approximation is less than ...
3
votes
2answers
145 views

What is the 35th derivative of $f(x) = e^{x^{10}} $at $x = 0$?

I had this question on a quiz and I answered 35!x^5/4! What is the 35th derivative of $f(x) = e^{x^{10}} $at $x = 0$? Use a suitable Taylor Polynomial for $e^x$ at $x = 0$. Express the answer in ...
3
votes
2answers
76 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 - ...
3
votes
4answers
75 views

Taylor series of $(1+x)\ln(1+x)$ in $x=0$

How to determine the Taylor series of $(1+x)\ln(1+x)$ in $x=0$? My idea is finding the second derivative of the expression, which is $\frac{1}{1+x}$. The Taylor series of this expression is ...
3
votes
3answers
311 views

Find an accurate value of $f(x)=\sqrt{4x^2+x}-2x$ for large values of x. Calculate $\lim_{x\to\infty}f(x)$

My works: $x^2$ can be very large if x is large, thus the function has lose-of-significance error and we need to reformulate it. $$ ...
3
votes
2answers
5k views

What is the Maclaurin series expansion for $\sqrt{x}$?

The derivative of $\sqrt{x}$ doesn't have a defined value at x = 0. How then do I find its maclaurin series expansion? Or can it only be approximated with a Taylor series at some value x != 0?
3
votes
3answers
35 views

Calculating using Taylor's series with Remainder: $ \lim \limits_{x \to 1} \frac{\ln x}{x^2+x-2} $

How to Calculate with Taylor's series with Remainder: $$ \lim \limits_{x \to 1} \frac{\ln x}{x^2+x-2} $$ without using L'Hopital's Rule? Here is what I reached: $$\lim \limits_{x \to 1} \frac{(x-1) ...
3
votes
3answers
232 views

Limit of $\dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ as $x \rightarrow 0$

Find $\lim_{x \to 0} \dfrac{\tan^{-1}(\sin^{-1}(x))-\sin^{-1}(\tan^{-1}(x))}{\tan(\sin(x))-\sin(\tan(x))}$ I came across this limit a long time ago and could easily obtain a straightforward solution ...
3
votes
2answers
73 views

Coincidence of $x-\frac{x^3}{6}$ and $\sin x$ in an interval

Plotting $f(x)=x-\frac{x^3}{6}$ and $g(x)=\sin x$, one can see that these two function are coincide in an interval $I\subset(-\frac{\pi}{2},\frac{\pi}{2})$. On the other hand, Taylor series for $\sin ...
3
votes
4answers
236 views

Trigonometric Coincidence

I Know that using Taylor Series, the formula of $\sin x$ is $$x-x^3/3!+x^5/5!-x^7/7!\cdots,$$ and the unit of $x$ is radian (where $\pi/2$ is right angle). However, the ratio of the circumference ...
3
votes
2answers
94 views

How can we determine the number of terms which we have to take in a series to get a particular accurate?

As I remember , two days ago , there was a question ( here ) asks for calculating this limit $\displaystyle \lim \limits_{x\rightarrow \infty } \frac{x^3}{e^x}$ and the question was answered . of ...
3
votes
3answers
761 views

Finding the Taylor series expansion of $f(z)=\frac{e^{z}-1}{z}$ around $0$

Find the Taylor series expansion of $f(z)=\displaystyle\frac{e^{z}-1}{z}$ around $0$. I have no idea where to start.
3
votes
2answers
214 views

Expanding logarithm into series

How to expand $f(x)=\ln\left( x+\sqrt{1+x^2} \right)$ into series at $x_0=0$? I've tried using Taylor's formula but counting consecutive derivatives was inconvenient and I couldn't find the general ...
3
votes
2answers
63 views

Finding $\lim_{x\to0} \frac{\sin(x^2)}{\sin^2(x)}$ with Taylor series

Evaluate $$\lim_{x\to0} \frac{\sin(x^2)}{\sin^2(x)}.$$ Using L'Hospital twice, I found this limit to be $1$. However, since the Taylor series expansions of $\sin(x^2)$ and $\sin^2(x)$ tell us that ...
3
votes
2answers
146 views

Proving $\forall x\in\mathbb R : \dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$ with Cauchy's MVT

Prove for all $x\in\mathbb R$: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ Mclauren expansion: $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+R_4(x)$$ ...
3
votes
4answers
93 views

Taylor Series of $ \frac{1}{1-x^2} $ about x=2

I am trying to form a taylor series of the following: $ \frac{1}{1-x^2} $ about $x=2$ I tried factoring the equation such that it becomes the following: $ \frac{1}{{(1+x)}{(1-x)}} $ I tried to ...
3
votes
2answers
158 views

The integral $\int\ln(x-\ln(x))~dx$

The integral $f(y)=\int_0^y\ln(x-\ln(x))~dx$ is on my mind. I'm not sure if this has a closed form? Maybe we need to use the lambert-W function to solve this one? If it cannot be done in closed ...
3
votes
1answer
81 views

Find all $\alpha \in \mathbb{R}$ such that $\int_0^\infty \sin(x^\alpha)dx$ converges.

Find all $\alpha \in \mathbb{R}$ such that $\int_0^\infty \sin(x^\alpha)dx$ converges. There is an answer here that differs from mine (they claim for $-\infty<\alpha<-2$ and ...
3
votes
3answers
861 views

Trigonometric identities using $\sin x$ and $\cos x$ definition as infinite series

Can someone show the way to proof that $$\cos(x+y) = \cos x\cdot\cos y - \sin x\cdot\sin y$$ and $$\cos^2x+\sin^2 x = 1$$ using the definition of $\sin x$ and $\cos x$ with infinite series. thanks...
3
votes
3answers
760 views

Remembering Taylor series

Could anyone suggest a good way of memorizing Taylor series for common functions? I have tried to remember them but never seem to be able to commit them to permanent memory.
3
votes
1answer
58 views

manipulations with Taylor expansions for log and sinh

How could we derive the equality $$ \frac14 \sum_{m=1}^\infty \frac1m \frac{1}{\sinh^2 \frac{m\alpha}2} = - \sum_{n=1}^\infty n\log (1-q^n)$$ where $q=e^{-\alpha}$ ?
3
votes
2answers
100 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
3
votes
2answers
237 views

Approximating the cosine by Taylor polynomial

Let $f:=\cos(x)$ I'm asked to find for which values of $x$ we can be sure the 4th degree Taylor polynomial will give an error lesser than $\frac{1}{1000}$. Now, ...
3
votes
5answers
261 views

Why don't taylor series represent the entire function?

Say, I have a continuos function that is infinitely differentiate on the interval $I$. It can then be written as a taylor series. However, taylor series aren't always completely equal to the function ...
3
votes
2answers
62 views

Maclaurin series of: $ f(x) = {x + 5\over1-x^2}$.

I'm trying to get the Maclauren series of: $ f(x) = {x + 5\over1-x^2}$. I am sure there is some trick here, the result according to Mathematica is: $5 + x + 5x^2 + x^3 + 5x^4 + x^5 + 5x^6 + \ ...$ ...