The graph shows the successive asymptotic approximations to the Stieltjes integr

Question

The graph shows the successive asymptotic approximations to the Stieltjes integral function S(x). Note that the more terms of the series included, the better the appoximation near x=0, yet the approximation pulls away from S(x) sooner. terms we add gets us clearer and closer to the function. For an asymptotic series, we first pick a number of terms, and we have an accurate approximation to the function, getting more accurate as x approaches a. Hence, asymptotic series can be used to compute complicated limits, regardless of whether the series converges or diverges. For complicated limits in which there is cancellation, we can replace a function with not just a similar function, but with the first several terms of its asymptotic series. Note that in figure 1. 4, the curves y = 1 and y = 1 - x + 2x2 cross at x = 1 / 2, the curves y = 1 - x and y = 1-x+2x2-6x3 cross at x = 1 / 3, and the curves y = 1 - x + 2x2 and 1 - x + 2x2 - 6x3 + 24x4 cross at x = 1/4. Show that the pattern continues. That is, show that the nth degree polynomial approximation and the (n - 2)nd degree polynomial approximation cross at x - 1/n.

Explanation / Answer

From the definition of the given integral function, it can be seen in the graph that the given property holds perfectly. It is the sole property of the the integral function S(x).

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