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.