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Expand-ln-1-sinh-x-as-a-series-in-ascending-powers-of-x-up-to-and-including-the-term-in-x-3-Hence-show-that-1-sinh-x-3-x-e-2-1-x-x-2-2-




Question Number 83295 by Rio Michael last updated on 29/Feb/20
Expand ln (1 + sinh x) as a series in  ascending powers of x up to and including  the term in x^3  . Hence , show that    (1 + sinh x)^(3/x)  ≅ e^2 (1 −x + (x^2 /2))
$$\mathrm{Expand}\:\mathrm{ln}\:\left(\mathrm{1}\:+\:\mathrm{sinh}\:{x}\right)\:\mathrm{as}\:\mathrm{a}\:\mathrm{series}\:\mathrm{in} \\ $$$$\mathrm{ascending}\:\mathrm{powers}\:\mathrm{of}\:{x}\:\mathrm{up}\:\mathrm{to}\:\mathrm{and}\:\mathrm{including} \\ $$$$\mathrm{the}\:\mathrm{term}\:\mathrm{in}\:{x}^{\mathrm{3}} \:.\:\mathrm{Hence}\:,\:\mathrm{show}\:\mathrm{that}\: \\ $$$$\:\left(\mathrm{1}\:+\:\mathrm{sinh}\:{x}\right)^{\frac{\mathrm{3}}{{x}}} \:\cong\:{e}^{\mathrm{2}} \left(\mathrm{1}\:−{x}\:+\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right) \\ $$

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