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Question Number 33348 by caravan msup abdo. last updated on 14/Apr/18
$${let}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}}\:{and} \\ $$$$\underset{{n}} {{T}}\:=\:\sum_{{k}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{\mathrm{2}{k}−\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{lim}\:{S}_{{n}} \:\:{and}\:{lim}\:{T}_{{n}} \left({n}\rightarrow\infty\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\Sigma\left({S}_{{n}} −{ln}\mathrm{2}\right)\:{and} \\ $$$$\Sigma\left({T}_{{n}} \:−\frac{\pi}{\mathrm{4}}\right){converges}\:{and}\:{find}\:{its}\:{sum} \\ $$$$ \\ $$