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

Commented by EDWIN88 last updated on 22/Mar/21

Use the fact that csc x = (1/x)+(x/6)+((7x^3 )/(360))+((31x^5 )/(15120))+... for ∣x∣ < π to  find the first four terms of the series for  2 csc (2x) [ for ∣x∣ < (π/2) and thus for ln (tan x).  (a) ln ∣x∣ +(x^2 /(12))+((7x^4 )/(1440))+((31x^6 )/(90720))+...  (b) ln ∣x∣−(x^2 /3)+((7x^4 )/(90))−((62x^6 )/(2835))+...  (c) ln ∣x∣ +(x^2 /3)+((7x^4 )/(90))+((62x^6 )/(2835))+...  (d) ln ∣x∣ −(x^2 /(12))+((7x^4 )/(1440))−((31x^6 )/(90720))+...

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

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