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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1answer
18 views

Determine radius of convergence of Taylor series of $f(z)$ at point $a$

Consider $$f(z) = \frac{z+e^z}{(z-1+i)(z^2-2)(z-3i)}, a=0 $$ As we can see it's quite ugly so I won't even try and develop a Taylor series of it at point $a=0$. I have noticed there are Four ...
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2answers
54 views

Sums of the series $1 + (x^2) / 3! +( x^4) / 5! +\cdots$

How can I compute sum of the series ; $$1 + \frac{x^2}{3!}+\frac{x^4}{5!}+\frac{x^6}{7!}+\frac{x^8}{9!}+\cdots$$ I tried to divide it to two pieces such that $$f(x) = ...
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2answers
46 views

Reindexing Exponential Generating Function

I have an exponential generating function, and I need to double check what the teacher said, because I'm having trouble coming to the same result. Also, I need to verify what I am coming up with, and ...
2
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2answers
65 views

Taylor's theorem with remainder of fractional order?

Let $k\geq 1$. Consider Taylor's theorem. We know the Peano form and the mean-value form of the remainder term: Peano form of the remainder Let $f\colon (-\varepsilon,\varepsilon)\to\mathbb R$ be ...
3
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0answers
58 views

A sufficient and necessary condition of Taylor series

Let $f(x)$ be a $C^{\infty}$ function on $(-R,R)$. Prove that $f(x)$ can be expanded as its Taylor series at the point $x=0$ over the interval $(-R,R)$ if and only if there exists a positive function ...
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2answers
53 views

Maclaurin series of $e^x\sin x$

Would you mind showing me a faster way of building Maclaurin series of $$f(x)=e^x\sin x$$ so I do not need to calculate a lot of derivatives?
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2answers
48 views

Maclaurin serie of $\frac{1}{(1-x)(1-2x)}$

Help me finding the Maclaurin serie of $$f(x) = \frac{1}{(1-x)(1-2x)} $$ in the easiest way (if there is one which you do not have to calculate a lot of derivatives) possible, please.
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1answer
80 views

Truncating a taylor expansion for a recurrence relation?

Let's say I have a function $N$ whose future value at a time $t + t_{d}$ obeys the relation $N(t + t_{d}) = A(t)N(t)$ where $A(t)$ is also a function of $t$ whose value can be calculated. One can ...
0
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1answer
18 views

How toexpress $V=\frac{kq}{x-a}-\frac{kq}{x+a}$ in terms of $k,q,x,u$ in Taylor Series for the following condition?

The question calls $u=\frac{a}{x}$ and $u$ is the variable. So for Taylor Series, we express it in $f(x)=\sum^{\infty}_{k=0}\frac{f^k(0)}{k!}x^k$ However, one hint says all we need is geometric ...
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0answers
36 views

How to find function $F$ such that $F''(x)=\cos{x^2}$, $F'(0)=3$ and $F(0)=4$?

Here we want $F\in \Bbb{R}$. We use Taylor Series. I get $F''(x)=\cos{x^2}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k+1)!}(x^2)^{2k}=\sum^\infty_{k=0}\frac{(-1)^k}{(2k+1)!}x^{4k}$ Integrating, we have ...
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0answers
28 views

Taylor Polynomial Approximtions

Answer Provided. Explanation needed. Hi, I am asked to construct a Taylor polynomial approximation that is accurate to within $10^{-3}$ over the indicated interval using $x_0=0$ with the following ...
2
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2answers
48 views

Marsden's definition of Taylor Series

How does the following definition of Taylor polynomials: $f(x_0 + h)= f(x_0) + f'(x_0)\cdot h + \frac{f''(x)}{2!}h^2+ ... +\frac{f^(k)(x_0)}{k!}\cdot h^k+R_k(x_0,h),$ where ...
0
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1answer
53 views

Determine the first four non-zero terms in the power series expansion about $x=0$ for the general solution: $\left(2x-3\right)y''-xy'+y=0$

$$\left(2x-3\right)y''-xy'+y=0$$ First I found the first to derivatives of the following power series: $$y(x)=\sum_{n=0}^{\infty}a_nx^n$$ $$y'(x)=\sum_{n=1}^{\infty}na_nx^{n-1}$$ ...
2
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1answer
29 views

Taylor Series, Approximation of a Function

a) Find the first 5 terms of the Taylor series for $f(x)=1/\sqrt{x}$ centered at $a=4$ b) use the result from a) to estimate $1/\sqrt{3}$ and compare it to calculated value My attempt at the ...
-1
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1answer
42 views

Compute the Taylor Series for $f\left(x\right)=\ln\left(1+x^2\right)$ about $x= 0$

I'm very confused by this question. Can you provide me with hints as to how to get started with this one? $f\left(x\right)=\ln\left(1+x^2\right)$ about $x= 0$ Do I just use the Taylor Series ...
3
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1answer
67 views

Finding the value of $1.1^{82}$ using $(1+x)^{82}$ to a certain accuracy

I found this question in a book. How many terms of the Maclaurin expansion of $(1+x)^{82}$ are needed to guarantee finding a value of $1.1^{82}$ to an accuracy of $10^{-6}$? This is how I tried to ...
0
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1answer
20 views

Taylor expanding to leading order

I've had a lot of trouble finding a reduced form of the solutions here to the leading order: $$\omega_{1,2}=-\frac{1}{2}(1+k+\epsilon) \pm \frac 12 \sqrt{(1+k+\epsilon)^2-4k\epsilon}$$ The textbook ...
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1answer
28 views

Converting this summation into an integral

This summation includes a sum of n derivatives of the function f(x) at the point (c+d) / 2 I'm trying to convert a Taylor ...
2
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2answers
51 views

Determine the first three non-zero terms in the Taylor polynomial approximation for the initial value problem: $y''+\sin(y)=0$

Having trouble understanding how to solve this problem. Did I at least set it up correctly? $y''+\sin(y)=0,\;y(0)=1,\;y'(0)=0$ So assuming $y(x)=\sum_{n=0}^{\infty}a_nx^n$ then ...
1
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1answer
22 views

propagation of error from product of Taylor Series

Say I have two functions $f(x)$ and $g(x)$, both of which I will be approximating with Taylor series $T_f(x)$ and $T_g(x)$ respectively. Lets say $f(x)$ is order $O(x^{n_1})$ and $T_f(x)$ has error of ...
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0answers
37 views

Derive the formula and its error from the basic rule $B(f: c,d)$

I understand Taylor series, for example I know the Taylor expansion for $f(a+bh)$ where $h \approx 0$ would be $$f(a+bh) = f(a) + bhf'(a) + b^2 h^2 f''(a)/2 + \cdots$$ But I have a problem that has ...
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1answer
26 views

Equivalence of Taylor series and its corresponding function and Axiom for infinite summation

Given a function $f(x)$ with a taylor series expansion, is it valid to say that $$f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}f^{(n)}(a)(x-a)^n$$ for all values of x irrespective of whether the taylor series ...
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3answers
26 views

show equality - binomial formula, taylor?

I am trying to show that this is true using the binomial formula or some taylor expansion: $\frac{1}{(1+\epsilon \sum\limits_{n=0}^\infty Z_n(t) \epsilon^n)^2} = 1 - 2Z_0\epsilon + ...
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1answer
22 views

Numerical analysis: what is the error term for the rule…?

The question goes: derive the error term for the rule $phi$ to approximate the third derivative of f(a). I have attached a screenshot I understand how to take the Taylor series in the hint, but the ...
1
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2answers
72 views

Estimating $\int_0^1f$ for an unknown Lipschitz $f$ to within 0.0001

A friend of mine has a Lipschitz function $f\colon [0,1]\to\mathbb R$ satisfying (Some more characters, and yet a few more...) $$|f(a)-f(b)| \le 5 |a-b| \qquad\text{for all }a,b\in[0,1],$$ ...
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4answers
3k views

On what interval does a Taylor series approximate (or equal?) its function?

Suppose I have a function $f$ that is infinitely differentiable on some interval $I$. When I construct a Taylor series $P$ for it, using some point $a$ in $I$, does $f(x) = P(x)$ for all $x$ in $I$? ...
4
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1answer
39 views

Maclaurin Series nth Derivative

Find $f^{(2016)}(0)$ if $f(x)=\sin(x^2)$. From the Maclaurin series, $$\sin(x^2)=\sum_{n=0}^\infty\frac{(-1)^nx^{4n+2}}{(2n+1)!}$$ Comparing coefficient, ...
0
votes
1answer
13 views

Radius of convergence work check

Original question Find radius of convergence of the Maclaurin series for $f(x)=(4-x)^{-0.5}$ Attemp at solution $$f(x)=\frac{1}{2}\sum_{n=0}^{\infty}\frac{(2n)!(x^n)}{2^{4n}(n!)^2}$$ ratio test: ...
1
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2answers
67 views

Quick Method to Calculate the Maclaurin Series of $\frac{1}{\sqrt{\cos{x}}} $

I am supposed to calculate the maclaurin series for $\frac{1}{\sqrt{\cos{x}}} $ but I can't seem to figure out an efficient way to go about doing this.
2
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0answers
63 views

Taylor series for multivalued complex functions (and their use in combinatorics)

As far as I know, it is considered to be a "fact" that by the Generalized Binomial Theorem, the complex function $\sqrt{1 + z}$ has the following Taylor expansion at $z = 0$: $$\sqrt{1 + z} = \sum_{n ...
11
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5answers
2k views

Why do un-integrable functions exist?

By un-integrable I mean functions whose antiderivative can not be expressed in terms of elementary functions. I recently learnt that any differentiable function can be expanded using the Taylor ...
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1answer
37 views

Matrix series convergence

Suppose we have the Maclaurin series of a function $f$, and it converges in a radius $R$. Then suppose we define a matrix argument to the function in a similar manner to the exponential definition of ...
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0answers
21 views

Taylor's Theorem for a function whose domain is $\mathbb R^n$

In my text book, Taylor theorem: Let $f:A\rightarrow\mathbb{R}$ be of class $C^{r}$ for $A\subset\mathbb{R}^{n}$ an open set. Let $x,y\in A$ , and suppose that the segment joining $x$ ...
1
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0answers
45 views

Taylor's formula and its quadratic term

I struggle with the following problem: For a function $$f: \mathbb{C} \rightarrow \mathbb{R}~,$$ $f$ attains its maximum for $z_0= e^{i\pi/3}$, $f(z_0)=F_{max}.$ Assume we may use Taylor's theorem ...
1
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1answer
32 views

Multivariable Taylor's Theorem [duplicate]

I chanced upon this lemma when studying differential geometry which seems to depend on Taylor's theorem, but I have never seen it before, could someone explain how the proof works? I am not sure how ...
2
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1answer
45 views

Find the Maclaurin series for $\ln(2-x)$

A little unsure if the result I got makes sense, so I want to ask here to be sure I am not doing something very silly. The Maclaurin series is given by ...
2
votes
2answers
61 views

find the value of $\int_0^1(C(-y-1)\cdot\sum_{k=1}^{2016}\frac{1}{y+k})dy$

The problem: find the value of $\int_0^1 (C(-y-1)\cdot\sum_{k=1}^{2016}\frac{1}{y+k}) dy$, where $C(\alpha$) is the coefficient of $x^{2016}$ in the Maclaurin series for $(1+x)^\alpha$ What I ...
2
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2answers
49 views

taylor expansion and limit of a series??

$f(x)=\int_0^xtan^{-1}tdt$ what is the taylor expansion about the origin of this function? and how do i use this to get the limit of the series $1-\frac{1}{2}-\frac {1}{3}+\frac {1}{4}+\frac ...
25
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9answers
14k views

Simplest proof of Taylor's theorem

I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem. By which I mean this: there are plenty of proofs that introduce some arbitrary construct: ...
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3answers
111 views

computing the series $\sum_{n=1}^\infty \frac{1}{n^2 2^n}$

$$\sum_{n=1}^\infty \frac{1}{n^2 2^n}$$ I am new in series thus I tried a pair of methods to compute but I couldn't
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0answers
22 views

Integral of Taylor for a Generic Function

I can't seem to figure out the solution to this problem: Problem: Find a series for the following equation and give the first 3 terms and the $nth$ term:$$g(x) = \int_3^xf(t)dt$$ The problem also ...
2
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2answers
103 views

Can we calculate $2^k$ using this easy Taylor series?

Trying to calculate $2^k$ by hand for $k\in[0,1]$, it's tempting to use the Taylor expansion of $x^k$ around $x=1$, to get: $$2^k = 1^k + \frac{k (1)^{k-1}}{1!} + \frac{k(k-1) (1)^{k-2}}{2!} + \ldots ...
11
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6answers
5k views

How do Taylor polynomials work to approximate functions?

I (sort of) understand what Taylor series do, they approximate a function that is infinitely differentiable. Well, first of all, what does infinitely differentiable mean? Does it mean that the ...
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0answers
50 views

Function $s(x)=1+\sum_{k=1}^{\infty} \frac{x^k}{k^k}$ - is there any other way to define it?

This series converges for all $x \in (-\infty, \infty)$, thus the function is analytic on the real line and defined by its Taylor series. However, unlike the exponential function, this one is very ...
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0answers
32 views

Asymptotic to $f^{-1}(f ' (x)) $?

Let $tr(n)$ be the triangular numbers and $te(n)$ be the tetrahedral numbers. $$g(x) := \sum \frac{x^n}{n! 2^{tr(n)}}$$ $g'(x) = g(\frac{x}{2}) $ Now consider the analogue $$ f(x) = \sum ...
1
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1answer
51 views

Error of Taylor Series?

Part of my assignment is to find the third degree Taylor Series of $\tan(x)$ about $\pi/4$ and then estimate the error of this approximation when evaluated at 0.75. Finding the series was easy ...
0
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1answer
628 views

Taylor expansion, integration by parts, and the integration of dt.

So my notes say, for a continuous function we have $$ \int_a^x f'(t)dt = f(x) - f(a) \tag 1 $$ which I understand. So re-arranging gives. $$ f(x) = f(a) + \int_a^x f'(t)dt \tag 2 $$ or $$ f(x) ...
0
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1answer
43 views

Other representation of the Lagrange remainder

so far I've only seen this representation of the Lagrange remainder $R_n=\dfrac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}$ for some $c$ between $x$ and $a$. $(i)$ Now I came across this representation: ...
0
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0answers
18 views

Differing notation in compact Taylor series for several variables

I'm a second year mathematical physics student. Wikipedia has a compact definition for the Taylor series in several variables: $$T(\textbf{x})=\sum_{\alpha\geq ...
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0answers
20 views

An approximation of denomiator

I am trying to figure out how to make an approximation of $\frac{1}{x^a+y}$ to separate the term $x$ and $y$? I have tried to use Taylor expansion, but it also left the same denominator term? Thanks ...