Question

Sum of each of the following infinite series ?

Answer 1

`7/4" and "23/35`

Explanation:

`(a)`

`"this is a "color(blue)"geometric series"`

`"with "a=3`

`r=a_2/a_1=(-3(5/7))/3=-5/7`

`S_oo=a/(1-r)=3/(12/7)=7/4`

`(b)`

`"these are also geometric sequences"`

`sum_(k=1)^oo((4/9)^k-(1/8)^k)`

`=sum_(k=1)^oo(4/9)^k-sum_(k=1)^oo(1/8)^k`

`"with " a=4/9" and "r=4/9" and "a=1/8/r=1/8`

`S_oo=(4/9)/(1-4/9)-(1/8)/(1-1/8)`

`color(white)(S_oo)=4/5-1/7=23/35`