To determine

To calculate: The sum of the series for the given partial sum.

Answer

The sum of series is 2.

Explanation

Given:

The partial sum is sn=2−3(0.8)n .

Result used:

(1) The sequence {rn} converges to zero when −1<r<1 .

(2) If limn→∞sn=L , then ∑n=1∞an=L .

Calculation:

Obtain the limit of the partial sum. (The value of the term sn as n tends to infinity).

limn→∞sn=limn→∞(2−3(0.8)n)=limn→∞(2)−limn→∞(3(0.8)n)

limn→∞sn=2−3⋅limn→∞(0.8)n (1)

Since −1<0.8<1 and by Result (1), the value of limn→∞(0.8)n is zero.

That is, limn→∞(0.8)n=0 .

Substitute 0 for limn→∞(0.8)n in equation (1),

limn→∞sn=2−3⋅0=2−0=2

Since limn→∞sn=2 and by Result (2) , ∑n=1∞an=2 .

Therefore, the sum of series is 2.