4. (a) Show that In(n+1)-Inn = - *** / dt - "* - dt = 7 dt 2+1 1 1 1 2 3 11 (b) Let a,,= 1+-+ ++ -Inn, for n=1,2,.... Us

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4 A Show That In N 1 Inn Dt Dt 7 Dt 2 1 1 1 1 2 3 11 B Let A 1 Inn For N 1 2 Us 1 (24.19 KiB) Viewed 84 times

4. (a) Show that In(n+1)-Inn = - *** / dt - "* - dt = 7 dt 2+1 1 1 1 2 3 11 (b) Let a,,= 1+-+ ++ -Inn, for n=1,2,.... Use (a) to show that (a) is a decreasing sequence. (Hint: Show that an an+1 20.) d dt with partition {1,2,..., n+1), show that 1+ (c) Using the left sum of In(n+1). (d) Use (c) and the definition of an to show that an 20 for all n. +1/² (e) Use (b) and (d) to show that (an) converges to a number r. (r≈ 0.577216, and is known as the Euler-Mascheroni constant).