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Question Number 197323 by Erico last updated on 13/Sep/23

$$\mathrm{If}\mathrm{f}\left(\mathrm{x}\right)=\frac{\sin \left(\mathrm{x}\right)}{\mathrm{x}}\mathrm{and}{\mathrm{S}}_{\mathrm{n}}\left(\alpha \right)=\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}[\mathrm{f}(\mathrm{k}\pi +\frac{\pi }{\alpha })+\mathrm{f}(\mathrm{k}\pi -\frac{\pi }{\alpha })](\alpha >1)$$$$\mathrm{Prove}\mathrm{that}\underset{\mathrm{n}\to +\mathrm{\infty }}{\lim }{\mathrm{S}}_{\mathrm{n}}\left(\alpha \right)=1-\mathrm{f}\left(\frac{\pi }{\alpha }\right)$$

Answered by witcher3 last updated on 14/Sep/23

$${\mathrm{S}}_{\mathrm{n}}\left(\mathrm{a}\right)=\underset{1}{\sum ^{\mathrm{n}}}(\mathrm{f}(\mathrm{k}\pi +\frac{\pi }{\mathrm{a}})-\mathrm{f}(\mathrm{k}\pi -\frac{\pi }{\mathrm{a}}))$$$$\mathrm{T}\left(\mathrm{a}\right)\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\frac{{(-1)}^{\mathrm{k}}\sin \left(\frac{\pi }{\mathrm{a}}\right)}{\mathrm{k}\pi +\frac{\pi }{\mathrm{a}}}$$$$-\mathrm{T}(-\mathrm{a})+\mathrm{T}\left(\mathrm{a}\right)={\mathrm{S}}_{\mathrm{n}}\left(\mathrm{a}\right)$$$$\mathrm{T}\left(\mathrm{a}\right)=\frac{\sin \left(\frac{\pi }{\mathrm{a}}\right)}{\pi }\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\frac{{(-1)}^{\mathrm{k}}}{\mathrm{k}+\frac{1}{\mathrm{a}}}...\mathrm{cv}$$$$\mathrm{\Sigma }{(-1)}^{\mathrm{k}}⩽2,..\frac{1}{\mathrm{k}+\frac{1}{\mathrm{a}}}\mathrm{decrease}\mathrm{cv}\to 0$$$$\mathrm{T}\left(\mathrm{a}\right)..\mathrm{cv}$$$$\underset{\mathrm{k}=1}{\sum ^{\mathrm{\infty }}}\frac{{(-1)}^{\mathrm{k}}}{\mathrm{k}+\frac{1}{\mathrm{a}}}=\underset{\mathrm{k}=1}{\sum ^{\mathrm{\infty }}}\frac{1}{2\mathrm{k}+\frac{1}{\mathrm{a}}}-\frac{1}{2\mathrm{k}-1+\frac{1}{\mathrm{a}}}$$$$=\frac{1}{2}\underset{\mathrm{k}=1}{\sum ^{\mathrm{\infty }}}\frac{1}{\mathrm{k}+\frac{1}{2\mathrm{a}}}-\frac{1}{\mathrm{k}+\frac{1}{2}(\frac{1}{\mathrm{a}}-1)}$$$$=\frac{1}{2}(\mathrm{\Psi }\left(\frac{1}{2}(\frac{1}{\mathrm{a}}+1)\right)-\mathrm{\Psi }(\frac{1}{2\mathrm{a}}+1))$$$${\mathrm{S}}_{\mathrm{n}}\left(\mathrm{a}\right)=\mathrm{\Gamma }\left(\mathrm{a}\right)-\mathrm{\Gamma }(-\mathrm{a})$$$$=\frac{\sin \left(\frac{\pi }{\mathrm{a}}\right)}{2\pi }(\mathrm{\Psi }(\frac{1}{2}+\frac{1}{2\mathrm{a}})-\mathrm{\Psi }(\frac{1}{2}-\frac{1}{2\mathrm{a}})-\mathrm{\Psi }(1+\frac{1}{2\mathrm{a}})+\mathrm{\Psi }(1-\frac{1}{2\mathrm{a}}))$$$$=\frac{\sin \left(\frac{\pi }{\mathrm{a}}\right)}{2\pi }(\mathrm{\Psi }(1-(\frac{1}{2}-\frac{1}{2\mathrm{a}})-\mathrm{\Psi }(\frac{1}{2}-\frac{1}{2\mathrm{a}})-2\mathrm{a}-\mathrm{\Psi }\left(\frac{1}{2\mathrm{a}}\right)+\mathrm{\Psi }(1-\frac{1}{2\mathrm{a}}))$$$$=\frac{\sin \left(\frac{\pi }{\mathrm{a}}\right)}{2\pi }(\pi \cot (\pi (\frac{1}{2}-\frac{1}{2\mathrm{a}})-2\mathrm{a}+\pi \cot \left(\frac{\pi }{2\mathrm{a}}\right)$$$$=\frac{\sin \left(\frac{\pi }{\mathrm{a}}\right)}{2}(\mathrm{tg}\left(\frac{1}{2\mathrm{a}}\right)+\cot \left(\frac{1}{2\mathrm{a}}\right))-\frac{\sin \left(\frac{\pi }{\mathrm{a}}\right)}{\frac{\pi }{\mathrm{a}}}$$$$=\frac{\sin \left(\frac{\pi }{\mathrm{a}}\right)}{2}\left(\frac{{\cos }^2\left(\frac{1}{2\mathrm{a}}\right)+{\sin }^2\left(\frac{1}{2\mathrm{a}}\right)}{\sin \left(\frac{\pi }{2\mathrm{a}}\right)\cos \left(\frac{\pi }{2\mathrm{a}}\right)}\right)-\mathrm{f}\left(\frac{\pi }{\mathrm{a}}\right)$$$$=1-\mathrm{f}\left(\frac{\pi }{\mathrm{a}}\right)$$$$$$

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