Answer:
The series is convergent answer ⇒ (a)
Explanation:
* The series is -8/5 + 32/25 + -128/125 + ........
- It is a geometric series with:
- first term a = -8/5 and common ratio r = 32/25 ÷ -8/5 = -4/5
* The difference between the convergent and divergent
in the geometric series is :
- If the geometric series is given by sum = a + a r + a r² + a r³ + ...
* Where a is the first term and r is the common ratio
* If |r| < 1 then the following geometric series converges to a / (1 - r).
- Where a/1 - r is the sum to infinity
* The proof is:
∵ S = a(1 - r^n)/(1 - r) ⇒ when IrI < 1 and n very large number
∴ r^n approach to zero
∴ S = a(1 - 0)/(1 - r) = a/(1 - r)
∴ S∞ = a/1 - r
* If |r| ≥ 1 then the above geometric series diverges
∵ r = -4/5
∴ IrI = 4/5
∴ IrI < 1
∴ The series is convergent