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
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 ...
2
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1answer
24 views

about truncated taylor expansions

I have a question about expanding $e^u$into a truncated taylor series where $u$ is itself a truncated Taylor series (in my example $u$ is expansion of $-\frac{\log(1+t)}{t}$, up to term $O(t^3)$), it ...
4
votes
2answers
203 views

Applying Taylor theorem on a linear map

I found the following in a stack of practice problems but had trouble dealing with it: Consider a linear map $A:C^\infty(\mathbb{R}^n)\rightarrow \mathbb{R}$ such that: If $f\in ...
0
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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
21 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 ...
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 ...
1
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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 ...
0
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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 ...
0
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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 ...
0
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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 + ...
0
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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...) $$|f(a)-f(b)| \le 5 |a-b| \qquad\text{for all }a,b\in[0,1],$$ ...
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
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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: ...
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 ...
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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 ...
2
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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 $x$ ...
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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
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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 ...
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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 ...
3
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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 ...
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 ...
0
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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
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 ...
4
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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
0
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2answers
28 views

Exponent of an Exponential Operator

There is a problem in my textbook that asks me to prove the following: For a bounded operator $A$ on a Hilbert space, prove that: $$(e^A)^n = e^{An} $$ for any natural number, $n$. However upon ...
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0answers
38 views

Find the error in the Maclaurin series for $\ln\left(\frac{1+x}{1-x}\right)$.

I have already that the series is, $$\ln\left(\frac{1+x}{1-x}\right)\approx 2\left(x+\frac{x^3}{3}+\frac{x^5}{5}+...+\frac{x^{2n+1}}{2n+1}\right).$$ The remainder is equal to, ...
0
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1answer
17 views

Trouble with integer partition proof

I am reading Keller & Trotter: Applied Combinatorics, pg. 155, and I am having trouble with an intermediate step in a proof. The proof deals with integer partitions: And the part I can't ...
0
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2answers
34 views

Expand $f(z)=\frac{1}{z^2(z-i)}$ in 2 different Laurent expansions around $z=i$ and tell where each converges.

My attempt: $$f(z)=\frac{1}{z^2(z-i)}$$ $$\frac{1}{z^2(z-i)}=\frac{Az+B}{z^2}+\frac{C}{(z-i)}$$ Solving for the unknown constants yields $$A=1$$ $$B=i$$ $$C=-1$$ Thus, $$f(z)=\frac{z+i}{z^2} - ...
1
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0answers
115 views

Taylor's series question on bounds

$f(x) = f(y) + f'(y)(x-y) + \frac{f"(y) (x-y)^2}{2} + ....$ is the taylor's series. I am aware of bounds which specify the error in approximating $f$ with a polynomial. I want sufficient conditions ...
2
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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 ...
0
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0answers
28 views

Help with taylor series summation?

$f(x) = \sqrt{x}$ When you do expansion, you get the following: $$4 + \frac{x-16}{2^3}\times 1! + \frac{(x-16)^2}{2^8}\times 2! + \frac{3(x-16)^3}{2^{13}}\times 3!$$$$+ ...
0
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1answer
46 views

Solving Second-order non-linear ODE, with fractional expansions

I am solving a differential equation related to fluid mechanics, a rigid air bubble rising towards the surface of a liquid. Doing all of the maths, I have come to this differential equation, which I ...
0
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0answers
30 views

Quality of $E(f(X))\approx f(EX)+\frac 1 2 f''(EX)\sigma_X^2$ approximation

For convex $f$, we have Jensen's lower bound $Ef(X)\ge f(EX)$. What conditions do we need to put on $f,X$ so that the second order expansion in the title would be an upper/lower bound/good ...
1
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1answer
25 views

Implicit Euler local error issue

I'm not sure I get what's going on here, and online resources are not helpful, at least I didn't find any helpful ones. For the problem: $$ \frac{dy}{dt} = f(t, y(t))$$ a numerical solution for ...
2
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0answers
25 views

An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
0
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0answers
94 views

Numerical Computation - Taylor series

I'm taking numerical computation course, and I have a problem with this question: Apply Taylor's formula to obtain a power series approximation about $a=0$ to $\sin(\pi x/2)$. Find the remainder, ...
1
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1answer
24 views

How to show $\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)}$

How to show the below equation ? $$\sum\limits_{i=1}^{t}\frac{1}{i}2^{t-i}=2^t\ln 2 -\frac{1}{2}\sum\limits_{k=0}^\infty \frac{1}{2^k(k+t+1)} ~~~~~(t\in \mathbb Z^+)$$
0
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1answer
38 views

Taylor Series Expansion of a function?

So, i was studying my Computer Vision lecture notes and i came across this formula which says Say, i have a function $f(x,y,t)$, $x,y$ and $t$ are the varying factors After $t+ \nabla t$, i have ...
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0answers
23 views

Loglinearize a nonlinear difference equation

There is an exercise in my weekly problem set I have to solve and I am really struggling. The setting is as follows: Suppose you have the following nonlinear difference equation in terms of Π_t and ...
1
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3answers
51 views

Find the laurent series for $\frac{1}{z(z-2)^2}$ centered at z=2 and specify the region in which it converges.

My attempt: $$\frac{1}{z(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{A}{z}+\frac{B}{z-2}+\frac{C}{(z-2)^2}$$ $$\frac{1}{z(z-2)^2} = \frac{(1/4)}{z}+\frac{(-1/4)}{z-2}+\frac{(1/2)}{(z-2)^2}$$ This is ...
1
vote
2answers
28 views

Taylor series of $\ln{\sqrt[4]{\frac{x-2}{5-x}}}$ to $o((x-x_0)^n)$ when $x_0 = 3$

Well I have tried to get it as $$f(x) = f(x_0) + \frac{f'(x_0)(x-x_0)}{1!} + \frac{f''(x_0)(x-x_0)^2}{2!} + ... + o((x-x_0)^n)$$ and got wrong results: First: $$f'(x) = \frac{3}{4(x-2)(5-x)}$$ ...
1
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1answer
45 views

Computing limits with Taylor series. [duplicate]

I'm posting my whole thought process, but I'd like to ask specifically about whether is my expansion of $\log(1+y)$ into Taylor series good & allowed in this situation? Is there an easier (not ...