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I tried to find what is the Taylor series of the function $\int_0^x \frac{\sin(t)}{t}dt .$

Any suggestions?

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    Sometimes "sinc" means this one $\mathrm{sinc}(x) = \sin(\pi x)/(\pi x)$.2013-01-09

2 Answers 2

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To settle this:

We have the Maclaurin expansion

$\sin\,t=t\left(1-\frac{t^2}{3!}+\frac{t^4}{5!}-\frac{t^6}{7!}+\cdots\right)$

Upon obtaining the expansion of $\dfrac{\sin\,t}{t}$ from this, integrate each term of this series expansion, using the formula

$\int_0^x t^k \mathrm dt=\frac{x^{k+1}}{k+1}$

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I just want to extend J.M.'s solution to a full solution of the exercise:

$\int_0^x \frac{\sin(t)}{t} \,dt$

Is with Maclaurin expansion ($\sin\,t=t\left(1-t^2/3!+t^4/5!-t^6/7!+\cdots\right)$) equals to

$\int_0^x \frac{t}{t}\left(1-\frac{t^2}{3!}+\frac{t^4}{5!}-\frac{t^6}{7!}+\cdots\right) \,dt = \int_0^x \left(1-\frac{t^2}{3!}+\frac{t^4}{5!}-\frac{t^6}{7!}+\cdots\right) \,dt $

We do now integrate this series to:

$\int_0^x \left(1-\frac{t^2}{3!}+\frac{t^4}{5!}-\frac{t^6}{7!}+\cdots\right) \,dt = x-\frac{x^3}{3 * 3!}+\frac{x^5}{5*5!}-\frac{x^7}{7 * 7!}+\cdots$

Above we did use the flowing fact: $\int_0^x t^k \, dt=\left.\frac{t^{k+1}}{k+1}\right|_0^x = \frac{x^{k+1}}{k+1}-\frac{0^{k+1}}{k+1} = \frac{x^{k+1}}{k+1}$

We can now write the result as a series: $x-\frac{x^3}{3 * 3!}+\frac{x^5}{5*5!}-\frac{x^7}{7 * 7!}+\cdots = \sum_{n=0}^\infty (-1)^n\frac{x^{(2n+1)}}{(2n+1) * (2n+1)!} = Si(x)$

And see that we get the Sine Integral.