Question

If `S_(n)=(1.2)/(3!)+(2.2^(2))/(4!)+(3.2^(2))/(5!)+...+` up to `n` terms, then sum of infinite terms is
A. `(4)/(pi)`
B. `(3)/(e)`
C. `(pi)/(r )`
D. `1`

Answer 1

Correct Answer - D
`(d)` Here `t_(r )=(r*2^(r ))/((r+2)!)=((r+2-2)2^(r ))/((r+2)!)=(2^(r ))/((r+1)!)-(2^(r+1))/((r+2)!)`
`:.S_(n)=sum_(r=1)^(n)t_(r )=sum_(r=1)^(n)(2^(r ))/((r+1)!)-(2^(r+1))/((r+2)!)=1-(2^(n+1))/((n+2)!)`
`:.S_(oo)=lim_(ntooo)(1-(2^(n+1))/((n+2)!))=1`