0
votes
1answer
28 views

Power series function expansion as solution for integral equation

I'm facing an integral equation whose unknown is a function $f(x)$: The equation is of the kind: $$ K = \int_{-l}^{l} G(x,s)f(s)ds $$ So it's a Fredholm integral equation that is rewritten in this ...
2
votes
2answers
56 views

Evaluate Definite Integral to desired accuracy

Evaluate $$\int_0^{1/2}x^3\arctan(x)\,dx$$ My work so far: $x^3\arctan(x) = \sum_{n=0}^\infty(-1)^n \dfrac{x^{2n+4}}{2n+1}$ $$\int_0^{1/2}x^3\arctan(x)\,dx = \sum_{n=0}^\infty ...
1
vote
0answers
63 views

Can there be a power series with interval of convergence $[k, \infty)$?

My answer : NO Because Interval of convergence is of the form $(a-R, a+R)$ Where $a$ is centre of convergence. If there exists a power series with Interval of convergence $[k, \infty)$ $ $ We ...
-1
votes
3answers
77 views

Can there be more than one power series expansion for a function.

I guess the answer is NO, for polynomials. I know that there are more than one series expansion for every function. But I am talking about power series here. All Ideas are appreciated
1
vote
2answers
111 views

Using series find $\int_0^1 \sqrt{1+x^4}\hspace{1mm} dx$ up to $2$ decimal places

I cannot figure out an aesthetic way to do this. Can someone give a beautiful solution to this ugly question? This is what I have tried yet. I used the fact that $$x = ...
2
votes
1answer
46 views

Showing integral on contour tends to zero

I'm trying to prove: $$\int \frac{e^{t(z+\frac{1}{z})}}{z^2} = \sum_0 ^{\infty} b_m t^{2m+1}$$ Where the integral is over a contour centre the origin, radius R, and the $b_m$ are some coefficients. ...
1
vote
1answer
37 views

Calculate the value of the integral of a series

let $$P(r,\varphi):= \dfrac{1}{2\pi} \sum_{n \in \mathbb{Z}} r^{|n|}e^{in\varphi} $$ with $\varphi \in \mathbb{R}$ and $ 0< r <1$. Prove that $$\int_{0}^{2\pi}P(r,\varphi)d\varphi =1$$ My ...
1
vote
0answers
39 views

Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
0
votes
1answer
66 views

Integrating a Taylor series term-by-term

Why is $$\int_{0}^{z} \frac{\sin x}{x} \ dx =\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!} \int_{0}^{z} x^{2n} \ dx$$ not valid for $z= \infty$? Well, at least I'm assuming it's not valid since ...
1
vote
1answer
64 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
3answers
230 views

Asymptotic expansion of $J(t) = \int^{\infty}_{0}{\exp(-t(x + 4/(x+1)))}\, dx$

I want to derive an asymptotic expansion for the following Bessel function. I think I need to rewrite it in another form, from which I can integrate it by parts. I am interested in obtaining the ...
1
vote
0answers
51 views

Equality between sum and integral

Show that,(where log is natural logarithm) $\lim_{n\rightarrow\infty}\left(logn\ -\sum_{k=0}^n\frac 1k\right)=\lim_{n\rightarrow\infty}\int\limits_{0}^{n}\left(1-\dfrac xn\right)^nlogx\ dx$ ...
0
votes
0answers
110 views

Given a power series, how do I find an integral representation of the function that the power series represents?

I have this power series: $$f(x) = \sum _{n=1}^{\infty } \frac{x n^{k n} (-x)^n (k!)^n}{(n+1) (k n)!}+x+1$$ I know that for $k=1$: $$f(x) = \text{x/LambertW(x)}$$ and that for $k=0$ and $x=1$ ...
3
votes
2answers
341 views

How to approximate the integral of the standard normal distribution.

So I have this eqn. $$ f(x)= \frac {e^ \frac{-x^2}{2}} {\sqrt{2\pi}} $$ I need to find: $$ \int\limits_{-1}^1 f(x)dx $$ So I want to use this series to integrate. I know that: $$ e^x = ...
11
votes
2answers
496 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
1answer
330 views

definite integral approximation using taylor series

In this question I cannot understand why they integrate to get the function that the power series represents...I also don't get how they selected the uppser and lower bound on the definite integral. ...
1
vote
1answer
75 views

Taylor expansion with integral?

I have looked at a version of a Taylor expansion that has an integral- for the first time. Is this the same as the usual version of a Taylor expansion without integrals? Also, do the $\alpha's$ have ...
1
vote
2answers
74 views

If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero

If $f$ is the sum of a convergent power series on a disk $D(0;R)$ prove that the integral of $f$ over any closed path $\gamma$ in $D(0;R)$ is zero. How can I able to prove the above problem without ...
2
votes
1answer
314 views

Evaluate a complex integral using power series expansions

Using power series expansions, evaluate the integral $$\int_{\gamma_r}\sin\left(\frac{1}{z}\right)dz.$$ where $\gamma_r:[0,2\pi]\rightarrow \mathbb C$ is given by $\gamma_r(t)=r(\cos t + i\sin ...
3
votes
3answers
118 views

What is the significance of this theorem (coefficients of power series as integrals)?

Isn't it easier to obtain the series' coefficicients by differentiation rather than by integration? The above text uses this theorem as an intermediate step in obtaining the generalised Cauchy ...
3
votes
0answers
302 views

Under which hypotheses is switching between sum and integral signs legit?

Which hypotheses are needed to change the order of sum and integral signs? Concrete example: consider the expression $$ ...
1
vote
0answers
613 views

Using power series to evaluate the integral of a piecewise function

I was self-studying Calculus, and the book I'm using asked me to solve the following integral using the technique of integration of power series: $$\int_{0}^{0.25} g(x)dx \text{ where } ...
2
votes
0answers
378 views

Approximation of integral using series expansion of the integrand.

I have a smooth function $x \rightarrow f_\epsilon (x)$ on $x\in[-1\ldots 1]$ (dependent on the continuous parameter $\epsilon$) and I want to approximate the integral $$ I=\int_{-1}^1 f_\epsilon ...
21
votes
3answers
504 views

Math contest: Find number of roots of $F(x)=\frac{n}{2}$ involving a strange integral.

Edit summary: A good answer appeared. CW full answer added, based on given answers. Removing my ugly-looking attempts, as they still remain in the rev. history. Here's a final-round calculus ...
4
votes
1answer
246 views

Definite integral of tetration between $0$ and $1$

In my old writes I found next formula, where is ${_{}^2}x$ is tetration: $$\int_0^1 {_{}^2}x \ dx = \sum\limits_{i=1}^\infty \frac {(-1)^{i+1}} {{_{}^2}i} \approx 0.783430511\ldots$$ And now I am ...
1
vote
1answer
191 views

simple pendulum

I have to deal with this integral in order to compute the period of a pendulum $$ \int^{\theta_{0}}_{0}\frac{d\theta}{\sqrt{\cos\theta_{0}-\cos\theta}} $$ I was asked by my instructor to solve this ...
1
vote
0answers
42 views

What, in general, can I expect to be the restrictions and/or limitations to this alternative process to rewriting?

This is a specific idea I have to rewriting $x$ as $x y$, which I recently asked about in this question. Suppose we have a power series $$f(x) = \sum_{i=0}^\infty{c_i x^i} = c_0 x^0 + c_1 x^1 + c_2 ...
2
votes
0answers
157 views

Confusion! Power series and integration

Consider the below power series: $\sum\limits_{n=1}^\infty \dfrac{x^{n}}{n^{2}}$ I know that it converges for $x\in [-1,1]$ and the sum $s(x)$ of the series is given by: $s(x) = - ...
1
vote
2answers
113 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 ...
5
votes
1answer
251 views

The net signed area between $t=0, y=0, t=x$, and $y=f(t)$

f(t) is continuous function.So I know that $\int _0^x {f(t) dt}=$ "The net signed area between $t=0, y=0, t=x$, and $y=f(t)$" And I can find the same result with endless small rectangulars areas ...
6
votes
3answers
375 views

Expressing the area of the image of a holomorphic function by the coefficients of its expansion

I have the following problem. Let $f:D\to \mathbb C$ be a holomorphic function, where $D=\{z:|z|\leq 1\}.$ Let $$f(z)=\sum_{n=0}^\infty c_nz^n.$$ Let $l_2(A)$ denote the Lebesgue measure of a set ...
4
votes
2answers
222 views

Proving the equivalence of a sum and a double integral

Based on "Certain Subclass of Starlike Functions" journal by Chun-Yi and Shi-Qiong Zhou in 2007 (Science Direct), I found difficulties to understand the proof in Theorem 3 where they have verified: ...
3
votes
1answer
290 views

Finding power series representation of $ \int_0^{\frac{\pi }{2}} \frac{1}{\sqrt {1 - k^2\sin^2{x}}}\;{dx}$

I want to show that $\displaystyle \int_0^{\frac{\pi }{2}} \frac{1}{\sqrt {1 - k^2\sin^2{x}}}\;{dx} = \frac{\pi}{2}\sum_{n \ge 0}k^{2n}\left({\frac{{1 \cdot 3 \cdots \left( {2n - 1} \right)}} {{2 ...
3
votes
0answers
224 views

Series of nested double integrals

This is kind of a follow-up of my previous question. I'm investigating the following infinite series of nested two-dimensional integrals $$\sigma(t,t^\prime) = 1 - \int_{t^\prime}^t\mathrm dt_1 ...
3
votes
2answers
396 views

Series of nested integrals

I'm trying to calculate the following series of nested integrals with $\varepsilon(t)$ being a real function. $$\sigma = 1 + \int_{t_0}^t\mathrm dt_1 \, \varepsilon(t_1) + \int_{t_0}^t\mathrm dt_1 ...
28
votes
3answers
748 views

On calculating $\int_0^1\ln(1-x^2)\;{\mathrm dx}$ — where is the mistake?

I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. ...