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Question Number 40047 by abdo mathsup 649 cc last updated on 15/Jul/18
$${let}\:{S}_{{n}} \:=\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{give}\:{S}_{{n}} \:{interms}\:{of}\:{H}_{{n}} \\ $$$$\left.\mathrm{2}\right){find}\:{lim}_{{n}\rightarrow+\infty} {S}_{{n}} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{W}_{{n}} =\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{4}{k}^{\mathrm{2}} −\mathrm{1}} \\ $$$${find}\:{W}_{{n}} \:\:{interms}\:{of}\:{H}_{{n}} \\ $$$${calculate}\:{lim}_{{n}\rightarrow+\infty} \:{W}_{{n}} \\ $$