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Today's calculation of Integral 807

Source: 2012 Osaka University entrance exam/Science, 2nd exam

March 14, 2012
calculusintegrationlimitcalculus computations

Problem Statement

Define a sequence ana_n satisfying : a1=1,  an+1=nan2+n(an+1) (n=1, 2, 3, ⋯ ).a_1=1,\ \ a_{n+1}=\frac{na_n}{2+n(a_n+1)}\ (n=1,\ 2,\ 3,\ \cdots).
Find lim⁡m→∞m∑n=m+12man.\lim_{m\to\infty} m\sum_{n=m+1}^{2m} a_n.
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