Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 170199 by Mastermind last updated on 18/May/22

$$Findthefirstfourtermsoftheseries$$$$for{e}^{x}sinhx$$$$$$$$Mastermind$$

Answered by floor(10²Eta[1]) last updated on 18/May/22

$$\mathrm{sinhx}=\frac{{\mathrm{e}}^{\mathrm{x}}-{\mathrm{e}}^{-\mathrm{x}}}{2}$$$$\therefore {\mathrm{e}}^{\mathrm{x}}\mathrm{sinhx}=\frac{{\mathrm{e}}^{2\mathrm{x}}-1}{2}=\mathrm{f}\left(\mathrm{x}\right)$$$$\mathrm{f}\left(0\right)=0$$$${\mathrm{f}}^{\prime }\left(\mathrm{x}\right)={\mathrm{e}}^{2\mathrm{x}}\Rightarrow {\mathrm{f}}^{\prime }\left(0\right)=1$$$${\mathrm{f}}^{″}\left(\mathrm{x}\right)={2\mathrm{e}}^{2\mathrm{x}}\Rightarrow {\mathrm{f}}^{″}\left(0\right)=2$$$${\mathrm{f}}^{‴}\left(\mathrm{x}\right)={4\mathrm{e}}^{2\mathrm{x}}\Rightarrow {\mathrm{f}}^{‴}\left(0\right)=4$$$$\Rightarrow {\mathrm{f}}^{\left(\mathrm{n}\right)}\left(\mathrm{x}\right)=2^{\mathrm{n}-1}{\mathrm{e}}^{2\mathrm{x}}\Rightarrow {\mathrm{f}}^{\left(\mathrm{n}\right)}\left(0\right)=2^{\mathrm{n}-1},\mathrm{n}>0$$$$$$$$\mathrm{f}\left(\mathrm{x}\right)=\underset{\mathrm{n}=0}{\sum ^{\mathrm{\infty }}}\frac{{\mathrm{f}}^{\left(\mathrm{n}\right)}\left(0\right){\mathrm{x}}^{\mathrm{n}}}{\mathrm{n}!}=\mathrm{x}+\frac{{2\mathrm{x}}^2}{2!}+\frac{{4\mathrm{x}}^3}{3!}+\frac{{8\mathrm{x}}^4}{4!}+...+\frac{2^{\mathrm{k}-1}{\mathrm{x}}^{\mathrm{k}}}{\mathrm{k}!}+...$$$$$$$$$$

Commented by Mastermind last updated on 19/May/22

$$thanks$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com