Could anybody help me with the following three questions, please?
They are in the topic of "the convergence of Darbox Sums and Riemann
Sums", "Mean Value Theorem", approximation of integrals, and Taylor
polynomials.
1. Let k >= 1 be an integer, and define Cn = SIGMA (1/(n+i )), where i=1 to kn
(a) Prove that {Cn} converges by showing it is monotonic and bounded.
(b) Evaluate LIMIT (Cn) AS n approach to the infinity
2. Let f(x) be integrable on [a,b], and let g(x) be nondecreasing and
continuously differentiable on [a,b]. Let {p be an element of P} be a
partition of [a,b], and define
U(f,g,p) = SIGMA (Mi(g(the ith term of x) - g(the (i-1) term of x))) where i=1 to n
L(f,g,p) = SIGMA (Ni(g(the ith term of x) - g(the (i-1) term of x))) where i=1 to n
Use mean value theorem to prove that (inf U(f,g,p), for p is element
of P) = (sup L(f,g,p), for p is the element of P) = ( INTEGRAL
f(x)g?(x)dx, where x is from a to b)
3.Given ( INTEGRAL ln square(x)dx, where x is from n to n+1 ) = (
INTEGRAL ln square (n+x)dx, where x is from 0 to 1 ) = ( INTEGRAL
[[ln(n+x) - ln(x) + ln(n)]square] dx, where x is 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)] |