Let
`C(theta)=sum_(n=0)^(oo) (cos(ntheta))/(n!)`
Which of the following statements is FALSE ?

Options

A. `C(theta).C(pi)=1`
B. `C(theta)+C(pi)gt2`
C. `C(theta)gt0"for all"thetainR`
D. `C^(')(theta)!=0"for all"thetainR`

Answer

D

Explanation

`C(theta)=underset(n=0)overset(oo)sum(cos(ntheta))/(n!)`
`=underset(ntooo)lim(1+(cos1theta)/(1!)+(cos2theta)/(2!)+(cos3theta)/(3!)+ . . .+(cosntheta)/(n!))`
`C(0)=1+(1)/(1!)+(1)/(2!)+(1)/(3!)+ . . . "up to"oo "term"`
=e
`C(pi)=1+(1)/(1!)+(1)/(2!)+(1)/(3!)+ . . . "up to"oo "term"`
`"Clearly" C(0).C(pi)=1`
`C(0).C(pi)=e+(1)/(e)gt2`
`C^(')=-(sintheta)/(1!)-2(sin2theta)/(2!)-(3sin3theta)/(3!)+ . . ."up to" oo "term"`
And that value is equal to zero at `theta=0`