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Today's calculation of Integral 717

Source: 1978 Osaka City University entrance exam

June 26, 2011
calculusintegrationgeometrylogarithmscalculus computations

Problem Statement

Let ana_n be the area of the part enclosed by the curve y=xn (n≥1)y=x^n\ (n\geq 1), the line x=12x=\frac 12 and the xx axis. Prove that : 0≤ln⁡2−12−(a1+a2+⋯⋯+an)≤12n+10\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}
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