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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)]

Solution Summary

Taylor polynomials are used to define limits.

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