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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10
votes
4answers
523 views

Show that $e^x \geq (3/2) x^2$ for all non-negative $x$

I am attempting to solve a two-part problem, posed in Buck's Advanced Calculus on page 153. It asks "Show that $e^x \geq \frac{3}{2}x^2$ $\forall x\geq 0$. Can $3/2$ be replaced by a larger ...
0
votes
1answer
31 views

Prove that the polynomial divided by a fraction of the power of n is equal to the sum of fractions of any constans and successive powers of

Let n≥1 and n is integer. P(x) - polynomial and $deg P(x)<n$. Prove if $ a \in \Bbb R/{0} $ then: $ \frac{P(x)}{(ax+b)^n} = \frac{c_1}{ax+b} + \frac{c_2}{(ax+b)^2}+...+\frac{c_n}{(ax+b)^n}$ for ...
2
votes
2answers
83 views

failed application of magicry in Taylor expansion of $1/x^2$ near $x=2$

It's straightforward to find the Taylor expansion for $\frac{1}{x^2}$ near $x=2$ using the the Taylor series definition. This is turns out to be $\frac{1}{4} - \frac{1}{4} (x-2) + \frac{3}{16}(x-2)^2 ...
0
votes
1answer
213 views

Confused about a limit proof and Big O.

I gave an incorrect proof here : How can evaluate $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$ I am confused as when considering the mistakes in my proof it seems the limit cannot be ...
0
votes
1answer
999 views

estimating the error of $\sin(x) = x$ with Taylor's Theorem

I want to calculate the numerical error in approximating $\sin(x)=x$ with Taylor's Theorem. Furthermore, what values of $x$ will this approximation be correct to within $7$ decimal places? Here is ...
0
votes
3answers
213 views

Taylor series of $ f = e^{x^2 + y^2}$ near $(0,0)$

I have to compute the second order Taylor series of the function $ f = e^{x^2 + y^2}$ near $(0,0)$. The Jacobian is: $$ Df(x,y) = (2\ x\ e^{x^2 + y^2}, 2\ y\ e^{x^2 + y^2}) $$ and the Hessian: $$ ...
8
votes
1answer
1k views

Taylor Series and Fourier Series

Taylor series expansion of function, $f$, is a vector in the vector space with basis: $\{(x-a)^0, (x-a)^1, (x-a)^3, \ldots, (x-a)^n, \ldots\}$. This vector space has a countably infinite dimension. ...
11
votes
2answers
510 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( ...
0
votes
0answers
38 views

How to Taylor expand $\ln{1-\exp{-i_t}}$ around i?

my question here is how to Taylor expand around $i$ $\ln{(1-\exp{(-i_t)})}$ to the first order? $i_t$ is a time series variable, $i$ is its steady state. Could anyone show me how to expand it ...
2
votes
0answers
59 views

Taylor polynomial of $\frac{1}{1-x-y}$

I need to calculate the 2nd order Taylor polynomial at the origin of $$f(x,y) = {1 \over{1-x-y}}$$ I have looked at two ways, and not sure which is simpler. We can split it by partial derivatives ...
0
votes
1answer
69 views

Differentiation term by term of Taylor series

Suppose I have A Taylor Series of a function around $z_{0}$ in the complex plane which convergence in a ball of radius $r>0$. Can I differentiate term by term the Taylor series and get the ...
0
votes
1answer
27 views

Taylor of $f:\Bbb R^3\to \Bbb R$

My notes say the following: You have a function $D(x, y, \sigma)$ mapping to a scaler. Take the taylor expension (which I can only do for functions from $\mathbb{R} \to \mathbb{R}$) up to the ...
1
vote
3answers
74 views

Find all the numbers $x$ such that $\sum_{n=0}^{\infty} \frac{x^n}{(2n)!}=0$

Find all the numbers $x$ such that $$\sum_{n=0}^{\infty} \frac{x^n}{(2n)!}=0$$ Is it by some tricks on Taylor series on $\sin{x}$, $e^x$?
1
vote
2answers
307 views

Determine whether a multi-variable limit exists $\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$

I need to determine whether the next limit exists: $$\lim_{(x,y)\to(0,0)}f(x,y)=\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$$ Looking at the numerator $(-1-\frac{x^2}2)$ it immediately ...
1
vote
1answer
86 views

trouble expanding taylor series about a point other than zero using geometric series

I'm trying to understand how to use a Taylor series expansion to correctly expand a population growth function about a point other than zero using the geometric series. For expansion about $t=0$, I ...
2
votes
1answer
97 views

Using Maclaurin series with solving a multi-variable limits

I need to determine wheter there's a limit where $(x,y)=(0,0)$ of the next function: $$\lim_{(x,y)\to(0,0)}\frac{e^{x(y+1)}-x-1}{\sqrt{x^2+y^2}}$$ In order to simplify the expression can I use ...
2
votes
1answer
123 views

Taylor Series Remainder

Use Taylor's Theorem to estimate the error in approximating $\sinh 2x$ by $2x + 4/3x^3$ on the interval $[-0.5,0.5]$. For this question, I use the Taylor's remainder formular, $$ R_n(x)= ...
0
votes
1answer
81 views

Expand log function with two terms

HOw can I expand ln(1+2/(A-1))? I think I need to use taylor series but the 1 is messing me up. Should I just ignore the 1?
0
votes
3answers
79 views

Taylor's Remainder

what is the maximum error when approximating $e^{x}$ by $1+x+\frac{x^{2}}{2}$ for $|x|<1$? Answer for this is $\frac{e}{6}$. Can anyone teach me the working for this question, please?
1
vote
2answers
39 views

How to evaluate binomial coefficients when $k=0$ and $1\geq|n|\geq0$

So I am doing some taylor series which boil down to the geometric series, for which I then need to evaluate various binomial coefficients. I have always used $n\choose k$ $=\frac{n!}{(n-k)!k!}$ to do ...
2
votes
1answer
81 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 ...
2
votes
1answer
68 views

Computing taylor series, getting all 0's

I started out by finding the first and second derivative. For $f'(x)$ I got $\;\;\dfrac{(12x^2-x^4)}{(4-x^2)^2}$ For $f''(x)$ I got $\;\;\dfrac{(4-x^2)(24x-4x^3)-(12x^2-x^4)(-4x) }{ (4-x^2)^3}$ ...
6
votes
3answers
95 views

Find $f^{(1001)}(0)$

I am to find the value in 0 of 1001th derivative of the function $$f(x) = \frac{1}{2+3x^2}$$ How should I approach this kind of problem? I tried something like : $$\frac{1}{2+3x^2} = ...
2
votes
2answers
204 views

What is the easiest/most efficient way to find the taylor series expansion of $e^{1-cos(x)}$ up to and including degrees of four?

So I have $$e^{1-cos(x)}$$ and want to find the taylor series expansions up to and including the fourth degree in the form of $$c_{0} \frac{x^0}{0!} + c_{1} \frac{x^1}{1!} + c_{2} \frac{x^2}{2!} + c_3 ...
0
votes
3answers
60 views

Calculate the sum $\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{n\times2^{2n+1}}$

I started with $arctg(x) = \sum\limits_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}$ Then I differentiated to get rid of the denominator. Then divide with $x$ to get $x^{2n-1}$. Then integrate to get ...
0
votes
1answer
49 views

why start the taylor series of $\cos^{2} x$ at $k=1$ and not just $k=0$ as I do not understand the problem with $2^{-1}$

Im using $\cos^2 x=\frac{1}{2}(1+\cos(2x))$ and $\cos x = (-1)^k \frac{(2x)^{2k}}{(2k)!}$ to find the sum for the Taylor series of $\cos^2 x$. I thought I was getting it. When I find the answer ...
2
votes
3answers
316 views

Why is domain of convergence of Taylor series of $\ln(x)$ about $x=1$ is $ (0,2)$?

I can understand the lower bound as $\ln(x)$ doesn't exist for $x<0$. But how is the upper bound $2$?
2
votes
3answers
74 views

Expand function into a Maclaurin's series

The function is given with: $f(x)=\dfrac{x^{2012}}{(1-x^3)^2}$ I have no idea what to do here and honestly, I don't really understand what a Maclaurin's series is. I know the definition but I don't ...
5
votes
4answers
83 views

Multiplying the long polynomials for $e^x$ and $e^y$ does not give me the long polynomial for $e^{x+y}$

As an alternative to normal rules for powers giving $e^xe^y=e^{(x+y)}$ I am multiplying the long polynomial of the taylor series of $e^x$ and $e^y$. I only take the first three terms: $$ ...
4
votes
2answers
90 views

Why can't you find all antiderivatives by integrating a power series?

if $f(x) = \sum\limits^{\infty}_{n=0}\frac{f^{(n)}(0)}{n!}x^n$ why can't you do the following to find a general solution $F(x) \equiv \int f(x)dx$ $F(x) = \int ...
4
votes
1answer
149 views

Question about a solution to a problem involving Taylor's theorem and local minimum

I've been studying "Berkeley Problems in Mathematics, Souza, Silva" and I came across this problem: Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Assume that $f(x)$ has a ...
1
vote
1answer
789 views

Maclaurin series for sin(x) representation

The Maclaurin series for $\sin(x)$ is: $$ \sin(x) = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} ... $$ Which according to wikipedia is: $$ \displaystyle \sum_{n=0}^{\infty} ...
3
votes
3answers
670 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...
1
vote
1answer
67 views

Estimate the scale of the power series with Poisson pdf-like terms

Sorry to bother you, but I guess that this question is not appropriate for MO, so I repost it here hoping that someone could give me a clue. I would like to have an estimate for the series $$P(t) = ...
3
votes
1answer
120 views

When computing the Taylor series of $(\cos x)^2$ how does the slide jump to concluding it is $1-(\sin x)^2$?

In the following slide it shows how the taylor series of $(\cos x)^2$ is computed: On the first line they simply take the taylor series of cosx and write it out twice, which makes sense. However, ...
2
votes
2answers
52 views

Other log solutions?

I am evaluating the expression: $\ln(1)$ And I know the trivial solution is $0$. Are there other solutions to this equation? I feel there should be, my logic is as follows: if: $\ln(1) = x ...
11
votes
4answers
724 views

Taylor expansion of $(1+x)^α$ to binomial series – why does the remainder term converge?

For $α ∈ ℝ$ the function $g_α \colon B_1(0) → ℝ, x ↦ (1+x)^α$ is $C^∞$ and $g_α^{(n)}(x) = n! \tbinom{α}{n}(1+x)^{α-n}$, where $\tbinom{α}{n} = \frac{α(α-1)\cdots(α-n+1)}{n!}$ is the generalized ...
3
votes
2answers
49 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 ...
2
votes
2answers
154 views

Taylor series expansion and approximation

I found this amazing question in the last calculus exam, but I don't know how to answer. Let $T(x) = \ln(1+a) + \frac{1}{1+a}(x-a) - \frac{1}{2(1+a)^2}(x-a)^2 +...+ ...
1
vote
1answer
34 views

Finding power series for $f(x) = \frac{4x+53}{x^2-x-30}$

Given $f(x) = \dfrac{4x+53}{x^2-x-30}$, display it as a power series and find the radius of convergence. then calculate $f^{(20)}(0)$ So what I did was look at the Taylor Series Formula: $$f(x) = ...
0
votes
2answers
93 views

Taylor series of $f(x)=\frac {e^x-1}{x}$

I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions. How to simplify the function so that it can be expanded more easily?
1
vote
1answer
59 views

Is $e^z\sum_{k=0}^\infty\frac{k^3}{3^k}z^k$ analytic inside $|z|=3$?

Am I correct that the following function is analytic at least inside $|z|=3$? (I used the ratio test.) The solutions manual says that the function is analytic on and inside |z|=1, so I wonder if I'm ...
1
vote
0answers
83 views

optimal control -Taylor expansion - PDE problem

I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point. For the given control ...
1
vote
0answers
171 views

Maclaurin series expansion of an expression that involves a fraction

In the context of statistical mechanics the "classical trace" is defined as $Tr(A e^{-\beta H}) = \int dr^N dp^N A e^{-\beta H}$ where $r^N$ and $p^N$ are phase space variables. So if $\Delta H$ is a ...
2
votes
3answers
223 views

Finding the Taylor series of $f(z)=\frac{1}{z}$ around $z=z_{0}$

I was asked the following (homework) question: For each (different but constant) $z_{0}\in G:=\{z\in\mathbb{C}:\, z\neq0$} find a power series $\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}$ whose sum ...
0
votes
2answers
102 views

Series expansion with remaining $log n$

I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant. $$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$ I'm trying to do a series ...
1
vote
3answers
74 views

Taylor Polynomial for $x^{1/3}$

a. Compute the Taylor polynomial $T_3(x)$ for the function $(x)^{1/3}$ around the point $x=1$. b. Compute an error bound for the above approximation at $x = 1.3$. I'm having trouble figuring ...
6
votes
4answers
2k views

Are Taylor series and power series the same “thing”?

I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general ...
1
vote
0answers
126 views

Taylor Expansion of Power Series

Suppose that $\space f:[0,1]\rightarrow \mathbb{R}$ is real analytic and that its power series expansion is: $\\ f(x)=\sum\limits_{n=0}^\infty a_nx^n$ Prove that there exists an $x_0\epsilon (0,x)$ ...
5
votes
1answer
242 views

Maclaurin series of $f(x)=\sinh(1/x)$?

As we know the formula of Maclaurin series for $f(x) = \sinh(x)$ is $f(x)=x+x^3/3! + x^5/5!+\ldots$ Could anyone tell me what is the Maclaurin series of $f(x)=\sinh(1/x)$?