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Approximation of Integrals and Taylor Polynomials

Given ( INTEGRAL ln square(x)dx, as x from n to n+1 ) = ( INTEGRAL ln square (n+x)dx, as x from 0 to 1 ) = ( INTEGRAL [[ln(n+x) - ln(x) + ln(n)]square] dx, as x from 0 to 1 ),

(a) Verify that ( LIMIT (n/ln(n)) [INTEGRAL (ln square (x)dx) - (ln square (n))] as n approach to the infinity ) = 1

(b) Compute LIMIT ((n square)/ln(n)) [ INTEGRAL ln square (x)dx - ln square (n) - ((ln(n))/n)]

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Well, before starting to solve the problem, I should mention that there is some vagueness in the question. From the description of the problem, I have figured out that I would use Taylor polynomials to approximate the integral. We have this information:

That I agree and there is no problem, but if you trust me, I will tell you that the following equation does not hold at all:

Upon your request, I can show you why, but for the moment let's forget about it. Of course, it is likely that there has been a typo.

Ok, it is possible, but time consuming to ...

Solution Summary

Taylor polynomials are used to define limits are given. The approximation of integrals and taylor polynomials are analyzed.