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Today's calculation of Integral 783

Source: 2012 Gakusyuin University entrance exam/Science

February 9, 2012
calculusintegrationcalculus computations

Problem Statement

Define a sequence a1=0, 11−an+1−11−an=2n+1 (n=1, 2, 3, ⋯ )a_1=0,\ \frac{1}{1-a_{n+1}}-\frac{1}{1-a_n}=2n+1\ (n=1,\ 2,\ 3,\ \cdots).
(1) Find ana_n.
(2) Let bk=k+1k (1−ak+1){b_k=\sqrt{\frac{k+1}{k}}\ (1-\sqrt{a_{k+1}}}) for k=1, 2, 3, ⋯k=1,\ 2,\ 3,\ \cdots.
Prove that ∑k=1nbk<2−1\sum_{k=1}^n b_k<\sqrt{2}-1 for each nn.
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