Sharp Estimates regarding the Remainder of the Alternating Harmonic Series
نویسنده
چکیده
In the present paper we obtain enhanced estimates regarding the remainder of the alternating harmonic series. More precisely, we show that 1 4n2 +a < ∣ ∣∣ ∣ ∣ n ∑ k=1 (−1)k−1 1 k − (−1)n−1 1 2n − ln2 ∣ ∣∣ ∣ ∣ 1 4n2 +b , for all n ∈N , with a = 2 and b = 2(3−4 ln2) 2 ln2−1 = 1.177398899 . . . . In addition, the constants a and b are the best possible with the above-mentioned property. Mathematics subject classification (2010): 11Y60, 40A05, 41A44, 33B15.
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تاریخ انتشار 2015