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Question Number 212359 by MrGaster last updated on 11/Oct/24

$$$$$$\underset{k=0}{\sum ^{\mathrm{\infty }}}(\frac{1}{\sqrt{4k+1}}-\frac{1}{\sqrt{4k+3}})$$

Answered by MathematicalUser2357 last updated on 13/Oct/24

$$0.622791226505......$$

Commented by Ghisom last updated on 14/Oct/24

$$\mathrm{exact}\mathrm{solution}\mathrm{possible}\mathrm{and}\mathrm{required}$$

Answered by Berbere last updated on 15/Oct/24

$$=\sum _{k⩾0}\frac{1}{2}(\frac{1}{\sqrt{k+\frac{1}{4}}}-\frac{1}{\sqrt{k+\frac{3}{4}}})=\frac{1}{2}(\zeta (\frac{1}{2},\frac{1}{4})-\zeta (\frac{1}{2},\frac{3}{4}))$$$$\zeta (s,z)=\sum _{n⩾0}\frac{1}{{(n+z)}^s}zetahurwitzfunction$$

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