Fundamental Theorem of Calculus

Fundamental Theorem of Calculus. Please see the attached PDF file. Define L : (0,1) ! R by L(x) = Z x 1 dt t a) Prove L is differentiable and strictly increasing on (0,1), with L0(x) = 1/x and L(1) = 0. b) Prove that L(x) ! 1 as x ! 1 and L(x) ! −1 as x ! 0+. (You may wish to prove L(2n) = Xn k=1 Z 2k 2k−1 dt t > Xn k=1 2−k(2k − 2k−1) = n 2 for all n 2 N) c) Using the fact that (xq)0 = qxq−1 for x > 0 and q 2 Q prove L(xq) = qL(x) for all q 2 Q and x > 0. d) Prove L(xy) = L(x) + L(y) for all x, y 2 (0,1) e) Let e = limn!1(1 + 1 n)n. Use L’Hopitals Rule to show that L(e) = 1. Since this problem is an analysis problem, please b... click for more

Limit

Limit. Please see the attached PDF file.

Real Analysis : Absolutely Integrable

Please see the attached PDF file. Prove that if f is absolutely integrable on [1,1), then lim n!1 Z 1 1 f(xn)dx = 0 Since this problem is an analysis problem, please be sure to be rigorous. 1

Infinite Series of Real Numbers (Cesaro summable)

Infinite Series of Real Numbers (Cesaro summable)

Infinite Series of Real Numbers

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