SHARP ESTIMATE OF THE REMAINDER OF SOME ALTERNATING SERIES

For any two real numbers α > 0 and β > −α , we show that the best constants a and b (the smallest a and the largest b) such that the inequalities ∞ 1 1 _{< k=∑n}+₁ (_α − _k 1 ₊ )^k−_β¹_< 2α n + a 2α n + b hold for every n > 1 are a = ⁽_α+¹_β − S(α, β ) ⁾⁻¹ − 2α and b = α + 2β, where S(α, β ) = _n∑=^∞₁⁽_α⁻_n¹₊⁾ⁿ⁻_β¹ . In particular, we recover the main result of [6] and answer a question, stated in [6], about the Gregory-Leibniz series _∑^{∞ (−}_2n¹₋⁾ⁿ⁻₁¹ . More precisely, we show that the best constants n=1 c and d such that the inequalities ∞ 1 1 < _k=∑n+₁⁽⁻_2k¹₋^)k−₁¹ < _4n + d 4n + c 4 hold for every n > 1 are c = − 4 and d = 0.