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(8CQ) Determine whether the series -5+25-125+... is convergent or divergent.

(8CQ) Determine whether the series -5+25-125+... is convergent or divergent.-example-1
User Nwaltham
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1 Answer

2 votes

Answer:

The answer is divergent ⇒ answer (b)

Explanation:

* The series is -5 + 25 + -125 + ........

- It is a geometric series with:

- first term a = -5 and common ratio r = 25/-5 = -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 = -5

∴ IrI = 5

∴ IrI > 1

∴ The series is divergent

User Linor
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