Question

Find the expansion of e^xSin(x)?

(a) \(e^{xSin(x)}=1+x^2-x^4/3+x^6/120-…\)

(b) \(e^{xSin(x)}=1+x^2+x^4/3+x^6/120+…\)

(c) \(e^{xSin(x)}=x+x^3/3+x^5/120+..\)

(d) \(e^{xSin(x)}=x+x^3/3-x^5/120+…\)

The question was posed to me in an interview.

The question is from Taylor Mclaurin Series in portion Differential Calculus of Engineering Mathematics

Answer 1

Right option is (b) \(e^{xSin(x)}=1+x^2+x^4/3+x^6/120+…\)

Easy explanation: Given, f(x) = e^xSin(x), f(0) = 1

Now, the expansion of xSin(x) is \(x^2-x^3/3!+x^6/5!+…\)

Hence, \(e^xSin(x)=e^y=1+y+y^2/2!+y^3/3!+…\)

Hence,

\(e^{xSin(x)}=1+(x^2-\frac{x^4}{3!}+\frac{x^6}{6!}+..)+\frac{(x^2-\frac{x^4}{3!}+x^6/6!+..)^2}{2!}+\frac{(x^2-\frac{x^4}{3!}+x^6/6!+…)^3}{6}+..\)

\(e^{(xSin(x))}=1+x^2-\frac{x^4}{3!}+\frac{x^6}{5!}+\frac{x^4}{2}-\frac{x^6}{6}+\frac{x^6}{6}+….\) (we neglect all other other terms by considering the options given)

Hence, \(e^{xSin(x)}=1+x^2+\frac{x^4}{3}+\frac{x^6}{120}+…\)