Question #d7237

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Answer 1 · 2017-03-21T19:52:12

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

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

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

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

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

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

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

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

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