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799

Part of 2012 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 799

Source: 2012 Tokyo Institute of Technology entrance exam, problem 4

2/29/2012
Let nn be positive integer. Define a sequence {ak}\{a_k\} by a1=1n(n+1), ak+1=−1k+n+1+nk∑i=1kai  (k=1, 2, 3, ⋯ ).a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).
(1) Find a2a_2 and a3a_3.
(2) Find the general term aka_k.
(3) Let bn=∑k=1nakb_n=\sum_{k=1}^n \sqrt{a_k}. Prove that lim⁡n→∞bn=ln⁡2\lim_{n\to\infty} b_n=\ln 2.
50 points
calculusintegrationlimitlogarithmsinductionstrong inductioncalculus computations