Question

Question #d7237

Answer 1

`color(blue)("Thus the sum must be converging")`
`color(blue)(lim_(n->oo) s=30xx5/3=50)`

`color(blue)(s=sum_(i=1to oo)30(2/5)^(i-1) )`

Explanation:

`a_1=30r^0 ->30r^(1-1)`

`a_2=30r^1->30r^(2-1)`

`a_3=30r^2->30r^(3-1)`

As `r=2/5` then `r^i` is decreasing as `i` is increasing

Thus `30r^i` is decreasing as `i` is increasing

`color(blue)("Thus the sum must be converging")`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Set the sum to be s

`s=30+30(2/5)+30(2/5)^2+30(2/5)^3+.. +30(2/5)^n`

`s(2/5)=30(2/5)+30(2/5)^2+..+30(2/5)^n+30(2/5)^(n+1)`

`s(1-2/5)=30-30(2/5)^(n+1) = 30[1-(2/5)^(n+1)]`

`s= 30xx[1-(2/5)^(n+1)]/(1-2/5)`

As this is an infinite series we have

`lim_(n->oo) s=30xx(1-0)/(3/5)`

`color(blue)(lim_(n->oo) s=30xx5/3=50)`

`color(blue)(s=sum_(i=1to oo)30(2/5)^(i-1) )`