# Tagged Questions

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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Equality between sum and integral

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