(1cq) determine whether the series -8/5+32/25-128/125+ is convergent or divergent.

The series is convergent answer ⇒ (a)

Step-by-step 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

b) 1/3 * 1/5

step-by-step explanation:

each smallest part is 1/15 since the written one is 1/5 and it's divided by 3 equal parts. (1/5 )/3 = 1/15 = 1/3 * 1/5