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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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
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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 ...
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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 ...
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0answers
47 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 $a=5$ ...
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2answers
420 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
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1answer
158 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 ...
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1answer
117 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 ...
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0answers
448 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
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2answers
455 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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5answers
120 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
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3answers
598 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?
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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 ...
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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.
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2answers
232 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
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2answers
101 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.
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2answers
58 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
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2answers
567 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 $\forall x \in \mathbb{R}-\{1\}$. ...
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2answers
445 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 $\operatorname{Log} (1 + e^z)$ ...
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3answers
115 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
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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 ...
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2answers
144 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 ...
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4answers
168 views

The even-numbered coefficients of the Maclaurin series of $ \frac{1}{\cos(x)} $ are odd integers.

Let’s consider $ G(z) \stackrel{\text{df}}{=} \dfrac{1}{\cos(z)} $ as the exponential generating function of the sequence of Euler numbers. How can one prove that in the Maclaurin series of $ G $, $$ ...
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2answers
72 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
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4answers
72 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 ...
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3answers
299 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. $$ ...
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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?
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3answers
34 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
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3answers
231 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
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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 ...
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4answers
235 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 ...
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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
742 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 ...
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2answers
62 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
145 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: $$e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+R_4(x)$$ ...
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4answers
92 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 ...
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votes
2answers
157 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
823 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
1answer
57 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 $q=e^{-\alpha}$ ?
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2answers
99 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
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2answers
221 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
233 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
58 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: $5 + x + 5x^2 + x^3 + 5x^4 + x^5 + 5x^6 + \ ...$ ...
3
votes
2answers
254 views

Is there a closed form expression for the Taylor series of exp((f(z))?

Given a holomorphic function $f(z) = \sum_{k=0}^\infty f_k z^k/k!$, is there a readable formula for the Taylor series of $\exp(f(z))$? Using the chain and product rules, one can obtain $$\partial_z ...
3
votes
1answer
468 views

Is the Taylor series comparable to Fourier series and spherical harmonics?

I am currently trying to grasp spherical harmonics and try to digest that we proved that the sine and cosine functions are a basis for the $L^2$ space of the squared-integrable functions. So as far ...
3
votes
3answers
385 views

Taylor polynomial of $f(x) = 1/(1+\cos x)$

I'm trying to solve a problem from a previous exam. Unfortunately there is no solution for this problem. So, the problem is: Calculate the Taylor polynommial (degree $4$) in $x_0 = 0$ of the ...
3
votes
3answers
669 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
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3answers
737 views

Prove that the Taylor series converges to $\ln(1+x)$.

Prove the following statement. For $0 \leq x \leq 1$, the Taylor Series, $\displaystyle x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ converges to $\ln(1+x)$ Any help will be greatly appreciated! ...
3
votes
1answer
122 views

Series around $s=1$ for an integral

Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral. What is the series expansion around $s=1$? ...