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How to prove that \[ \sum_{i=1}^{\infty}\frac{1}{2^{3i}}\left(\csc^{2}\left(\frac{\pi x}{2^{i}}\right)+1\right)\sec^{2}\left(\frac{\pi x}{2^{i}}\right)\sin^{2}\left(\pi x\right)=1 \] for all \( x\in\mathbb{R} \).

Using graph, we can see that the value of this series is 1 for all values of x.

[DESMOS]{"version":7,"graph":{"viewport":{"xmin":-2.7634114187202106,"ymin":-4.543149781969371,"xmax":17.23658858127979,"ymax":8.437278166514556}},"randomSeed":"2ce535cd8c2a6f2ce21bf7669d621391","expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"\\sum_{i=1}^{N}\\frac{1}{2^{3i}}\\left(\\csc^{2}\\left(\\frac{\\pi x}{2^{i}}\\right)+1\\right)\\sec^{2}\\left(\\frac{\\pi x}{2^{i}}\\right)\\sin^{2}\\left(\\pi x\\right)"},{"type":"text","id":"4","text":"N substitutes for infinity. Use slider to change value."},{"type":"expression","id":"3","color":"#388c46","latex":"N=9","hidden":true,"slider":{"hardMin":true,"hardMax":true,"min":"1","max":"20","step":"1"}},{"type":"expression","id":"2","color":"#2d70b3"}]}}[/DESMOS]

Using graph, we can see that the value of this series is 1 for all values of x.

[DESMOS]{"version":7,"graph":{"viewport":{"xmin":-2.7634114187202106,"ymin":-4.543149781969371,"xmax":17.23658858127979,"ymax":8.437278166514556}},"randomSeed":"2ce535cd8c2a6f2ce21bf7669d621391","expressions":{"list":[{"type":"expression","id":"1","color":"#c74440","latex":"\\sum_{i=1}^{N}\\frac{1}{2^{3i}}\\left(\\csc^{2}\\left(\\frac{\\pi x}{2^{i}}\\right)+1\\right)\\sec^{2}\\left(\\frac{\\pi x}{2^{i}}\\right)\\sin^{2}\\left(\\pi x\\right)"},{"type":"text","id":"4","text":"N substitutes for infinity. Use slider to change value."},{"type":"expression","id":"3","color":"#388c46","latex":"N=9","hidden":true,"slider":{"hardMin":true,"hardMax":true,"min":"1","max":"20","step":"1"}},{"type":"expression","id":"2","color":"#2d70b3"}]}}[/DESMOS]

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