3
votes
0answers
33 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ This could be used to evaluate partial sums using knowledge of the ...
1
vote
0answers
30 views

Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
3
votes
1answer
100 views

TaylorSeries of complete elliptic integral of the first kind

I want compute $K(k)$ as a Taylor Series; $k\in\mathbb{R}$ and $\vert k \vert < 1$. Can someone help me? $$ K(k):= \int^{\frac{\pi}{2}}_0 \dfrac{dt}{\sqrt{1-k^2 sin^2t}} $$ Results so far: $$ ...
0
votes
1answer
32 views

Maclaurin series and taylor

Im trying to find the first four terms of Maclaurin series of $\space0.15t^2$ and evaluate $$\int_0^1 e^{-0.15}t^2 dt $$ Please this is a revision question. How do i go about it?
3
votes
1answer
84 views

Taylor expansion of a not easily differentiable function

Context: I'm trying to find the period of a simple pendulum. As is well known, if the initial angle is small the period is approximately constant. I'm trying to do a second order expansion. I have ...
3
votes
2answers
82 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
3
votes
0answers
65 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
0
votes
1answer
51 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
4
votes
2answers
80 views

Taylor series of an integral function

Problem $$I(x) = \int_{1}^x \frac{e^t - 1}{t}$$ Find $I'( \sqrt{x} )$. Solution We know that $F'(x) = f(x)$ by the fundamental theorem of calculus so $$I'(x) = \frac{e^t -1}{t}$$ And so $$I'( ...
0
votes
1answer
52 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) ...
2
votes
3answers
63 views

standard Taylor series using substitution

Find Taylor series using substitution about $0$ for $f(x)=\frac{125}{(5+4x)^3}$ by writing $\frac{125}{(5+4x)^3}=\frac{1}{(1+\frac{4}{5}x)^3}$? Determine a range of validity for this series.
2
votes
1answer
90 views

How would I integrate $e^{e^x}$?

Is there a way to integrate: $e^{e^x}$ without using a Taylor or McLaurin Series expansion?
0
votes
1answer
45 views

Taylor series of a rational function

I am facing some complicated integral, which part of it is $$\frac{z^{M-1}}{(1+(\eta z)^n)^p}$$ I think if I find the taylor series of this part the integral might be solved. So, can someone help me ...
3
votes
0answers
311 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
0
votes
2answers
95 views

Maclaurin polynomial for $\arcsin(x)$

How would I find the 3rd-order Maclaurin polynomial for $f(x) = \arcsin(x)$; with the interval $(0,\frac 3 4)$ to show it in terms of $x$? Would you have to somehow manipulate it to $\dfrac{1}{1+x}$ ...
3
votes
1answer
77 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 ...
2
votes
1answer
84 views

Taylor series in order to find the approximate antiderivative of a function

Somewhat inspired by this question about antiderivatives, I started to check whether or not that function had an elementary antiderivative. Then, after checking with Maxima, it struck me that, by ...
1
vote
1answer
68 views

Maclaurin Series of $\int_0^x \cos t^2\,dt$

Find the Maclaurin Series for $\int_{0}^{x}\cos t^2\,dt$. $$\cos(x) = \sum\frac{(-1)^n x^{2n}}{2n!}$$ I'm trying this: $$\cos^2 x = \sum\frac{(-1)^n x^{4n}}{(2n!)^2}$$ How would you solve this ...
4
votes
2answers
203 views

Evaluating $\int_{0}^{\frac{\pi}{2}} \arctan( a \sin x) \ dx$ using the Taylor expansion of $\arctan (x)$

I was wondering if it's possible to show that for $a >0$, \begin{align}\int_{0}^{\pi/ 2} \arctan (a \sin x) dx &= 2 \sum_{k=0}^{\infty} \frac{\left(\frac{\,\sqrt{\vphantom{\Large A}\,1 + ...
0
votes
2answers
49 views

Version of Taylor: $F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h \right \rangle.$

My teacher claimed without proof that Taylor's theorem with remainder implied that for a suitable function $F: \mathbb{R}^n \to \mathbb{R}$, $$F(x+h)-F(x) = \left \langle \int_0^1 \nabla F(x+th)dt, h ...
1
vote
0answers
66 views

maclaurin polynomial upper limit

I have the following integral: $$\int_{0}^{1/2} e^{x^2}dx$$ i have approximated the 5th degree maclaurin polynomial of the integral to be: $1+x^2+(1/2)x^4$. I need to obtain an upper bound on the ...
6
votes
2answers
142 views

Gamma Type Integral

I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ...
0
votes
1answer
75 views

Taylor Expansion with Integral Remainder Question

I have the following question at hand and I have to admit that I am not used to integral remainder form of taylor approximation. I am still trying to work around, so a couple of hints would be useful ...
1
vote
0answers
55 views

Looking for examples where $f(z)=\operatorname{inv} \int_{0}^{z} g(z)\, dz$ with $f(z)$ entire and $g(z)$ not meromorphic.

I'm looking for examples where $f(z)=\operatorname{inv}\int_{0}^{z} g(z) \, dz$ with $f(z)$ entire and $g(z)$ not meromorphic. For clarity, by $\operatorname{inv}$, I mean the functional inverse. ...
3
votes
1answer
105 views

Differentiate a hypergeometric function expression

I have the following function $$f_\epsilon (p)=\frac{1}{2}(1-p)^\epsilon 2^\epsilon {_2}F_1(1-\epsilon,\epsilon;1+\epsilon;\frac{1-p}{2}),\qquad p\in(-1,1).$$ Here $F$ is the hypergeometric ...
11
votes
2answers
499 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( ...
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 ...
1
vote
1answer
121 views

Integral remainder converges to 0

I want to show that $\displaystyle \log(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}$ for $-1<x\leq1$ i want to show it with the integral remainder of the taylor series that gave me: ...
0
votes
2answers
323 views

taylor expansion of an integral $\int_0^1{e^{x^2}}$

I need to calculate $\int_0^1{e^{x^2}\:dx}$ with taylor expasin in accurancy of less than 0.001. The taylor expansion around $x_0=0$ is $e^{x^2}=1+x^2+\frac{x^4}{3!}+...$. I need to calculate when the ...
0
votes
1answer
75 views

Maclauren Series and taylor polynomials

Question: Suppose that the function $k(x)$ has a maclauren series that converges $\left(-\frac{1}{2} , \frac{1}{2}\right]$ and you are told that $|k^{(n)}(x)| \leq 10$ at all $|x| \leq ...
1
vote
0answers
45 views

Taylor expansion of an integral in spherical co-ordinates

I've some difficulty deriving this equation from jackson electrodynamics (The equation after 1.30) $\nabla^2 \Phi_a\left({\textbf{x}}\right)=-\frac{1}{\epsilon_0}\int_{0}^{R} ...
3
votes
1answer
72 views

What's incorrect with this Taylor series derivation?

Let's \begin{align} f(T)= &f(0)+ \int_0^T f' (t)dt\\ f(T)=&(0)+f' (T)T-\int_0^T f'' (t)tdt\\ f(T)=&(0)+f' (T)t-f'' (T) \frac{T^2}{2}+\int_0^Tf''' (t) \frac{t^2}{2} dt\\ f(T)=&f(0)+f' ...
1
vote
1answer
300 views

Integration on the unit sphere

I have an integral on the unit sphere as follows. $$I(\mathbf{s}_1, \mathbf{s}_2) = \int_{\mathbb{S}^2} f(\mathbf{x} \cdot \mathbf{s}_1)f(\mathbf{x}\cdot\mathbf{s}_2)d\mathbf{x} $$ where the ...
1
vote
1answer
88 views

Integration using Taylor approximations

I am stuck on two problems: 1) Prove that $$\int_0^1 \frac{1+x^{30}}{1+x^{60}}dx=1+\frac{c}{31}$$ where $0< c <1$. 2) Prove that ...
1
vote
2answers
305 views

Calculate the improper integral and the taylor series of $f(x) = \int_{x}^1 \frac{tx}{\sqrt{t^2-x^2}} \,dt$

For the given function $$f(x) = \int_{x}^1 \frac{tx}{\sqrt{t^2-x^2}} \,dt$$ with -1 < x < 1. Calculate the improper integral. Calculate the Taylor series of $f(x)$ at $x=0$ until the third ...
0
votes
2answers
157 views

Approximate with error bounds, the integral $ \int^1_0 \dfrac{\sin x}{x}\,dx $

I actually already have the solution to this, but would just like some clarification of how the solution was reached. The solutions provided used the fact that by Taylor's theorem, $\sin x = T_6(x) ...
2
votes
1answer
86 views

Equivalent of $ \int_1^x e^{-\sqrt{\ln(t)}} \mathrm dt $ when $x \rightarrow \infty$

How can I prove that: $$ \int_1^x e^{-\sqrt{\ln(t)}} \mathrm dt \sim_{x \rightarrow \infty} xe^{-\sqrt{\ln(x)}}$$ without using l'Hôpital's rule ? Integration by parts: $$ \int_1^x ...
1
vote
2answers
117 views

Problem regarding infinite sum of remainders.

Before here @math.SE there was a question regarding a problem on a maths magazine. I decided to look at the link provided, and one problem proposed was (if I'm not recalling this wrongly): Find ...
9
votes
1answer
497 views

The error term in Taylor series and convolution.

I've been wondering a lot why is the remainder of the Taylor expansion of a function, $R_n(x)$, expressed (in one of the many forms) as something very similar to aconvolution. Precisely: $$R_n(x) = ...
3
votes
1answer
118 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 $s=1$? ...
1
vote
1answer
667 views

Approximate the integral $\int_0^1 \sin(x^2) dx$

I'd like to ask if someone can please give me a little push with this assignment: Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations ...