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Question Number 136475 by EDWIN88 last updated on 22/Mar/21

Commented by EDWIN88 last updated on 22/Mar/21

$$\mathrm{Use}\mathrm{the}\mathrm{fact}\mathrm{that}\csc \mathrm{x}=\frac{1}{\mathrm{x}}+\frac{\mathrm{x}}{6}+\frac{{7\mathrm{x}}^3}{360}+\frac{{31\mathrm{x}}^5}{15120}+...\mathrm{for}\mid \mathrm{x}\mid <\pi \mathrm{to}$$$$\mathrm{find}\mathrm{the}\mathrm{first}\mathrm{four}\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{series}\mathrm{for}$$$$2\csc \left(2\mathrm{x}\right)[\mathrm{for}\mid \mathrm{x}\mid <\frac{\pi }{2}\mathrm{and}\mathrm{thus}\mathrm{for}\ln \left(\tan \mathrm{x}\right).$$$$\left(\mathrm{a}\right)\ln \mid \mathrm{x}\mid +\frac{{\mathrm{x}}^2}{12}+\frac{{7\mathrm{x}}^4}{1440}+\frac{{31\mathrm{x}}^6}{90720}+...$$$$\left(\mathrm{b}\right)\ln \mid \mathrm{x}\mid -\frac{{\mathrm{x}}^2}{3}+\frac{{7\mathrm{x}}^4}{90}-\frac{{62\mathrm{x}}^6}{2835}+...$$$$\left(\mathrm{c}\right)\ln \mid \mathrm{x}\mid +\frac{{\mathrm{x}}^2}{3}+\frac{{7\mathrm{x}}^4}{90}+\frac{{62\mathrm{x}}^6}{2835}+...$$$$\left(\mathrm{d}\right)\ln \mid \mathrm{x}\mid -\frac{{\mathrm{x}}^2}{12}+\frac{{7\mathrm{x}}^4}{1440}-\frac{{31\mathrm{x}}^6}{90720}+...$$

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