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Today's calculation of Integral 837

Source: 2012 Shiga University of Medical Science entrance exam

July 9, 2012
calculusintegrationlimitlogarithmscalculus computations

Problem Statement

Let fn(x)=∑k=1n(−1)k+1(x2k−12k−1+x2k2k).f_n(x)=\sum_{k=1}^n (-1)^{k+1} \left(\frac{x^{2k-1}}{2k-1}+\frac{x^{2k}}{2k}\right).
Find lim⁡n→∞fn(1).\lim_{n\to\infty} f_n(1).
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