Question

What is the coefficient of x^101729 in the series expansion of cos(sin(x))?

(a) 0

(b) ^1⁄101729!

(c) ^-1⁄101729!

(d) 1

This question was addressed to me in exam.

My question is based upon Taylor Mclaurin Series topic in section Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (a) 0

For explanation I would say: We know that the series expansion of cos(x) is

cos(t)=1-\(\frac{t^2}{2!}+\frac{t^4}{4!}….\infty\)

Now substituting t=sin(x) we have

cos(sin(x))=\(1-\frac{1}{2!}\times (\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}+…\infty)^2+\frac{1}{4!}\times (\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}+…\infty)^4+..\infty\)

Observe that every term has odd powered series raised to an even term.

Thus, we must have only even powered terms in the above series expansion. The coefficient of any odd powered term is zero.