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695

Part of 2011 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 695

Source: 2011 Tokyo Medical and Dental University entrance exam

5/7/2011
For a positive integer nn, let
Sn=∫011−(−x)n1+xdx,  Tn=∑k=1n(−1)k−1k(k+1)S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}
Answer the following questions:
(1) Show the following inequality.
∣Sn−∫0111+xdx∣≤1n+1\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}
(2) Express Tn−2SnT_n-2S_n in terms of nn.
(3) Find the limit lim⁡n→∞Tn.\lim_{n\to\infty} T_n.
calculusintegrationlimitlogarithmscalculus computations