The sum of the series `1/1.2-1/2.3+1/3.4-1/4.5+` ... is

Options

A. `(1)/(n+1)`
B. `1-(1)/(n+1)`
C. `(1)/(n+1)-1`
D. `1+(1)/(n+1)`

Answer

B

Explanation

Let `T_(r)" be the "r^(th)` term of the given series. Then,
`T_(r)=(1)/(r(r+1)),r=1,2, . . .,n`
`rArr" "T_(r)=(1)/(r)-(1)/(r+1),r=1,2, . . .,n`
Let S be the required sum. Then,
`S=underset(r=1)overset(n)sumT_(r)=underset(r=1)overset(n)sum((1)/(r)-(1)/(r+1))`
`rArr" "S=(1-(1)/(2))+((1)/(2)-(1)/(3))+((1)/(3)-(1)/(4))+ . . . . +((1)/(n)-(1)/(n+1))`
`rArr" "S=1-(1)/(n+1)`