Answer:
The given series converges and [tex]\lim_{n \to \infty} a_n =0[/tex]
Step-by-step explanation:
Given is 108, -18, 3,...
It is an alternate series. An alternate series is convergent if:
1. The series is decreasing.
2. If the last term (n-th term) of series converges to 0.
Since each next term is less than its preceding term, so it is decreasing.
First term, a = 108
Second term = -18
Common ratio, r = -18/108 = -1/6
General term, aₙ = a*rⁿ = 108*(-1/6)ⁿ
When n increases to infinity, the exponent term will decreases to zero, and last term (n-th term) will converge to 0 as well.
Hence, the given series converges.
Now limit will be: [tex]\lim_{n \to \infty} a_n[/tex]
[tex]\lim_{n \to \infty} 108*(-1/6)^n \\= 108*(-1/6)^\infty \\= 108*(1/\infty) \\= 108*0 \\= 0[/tex]
Hence, [tex]\lim_{n \to \infty} a_n =0[/tex]
