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3 years ago

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

Mathematics

1 answer:

Ivan3 years ago

3 0

Answer:

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

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Data given and notation n

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$p_v$ represent the p value (variable of interest)

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Concepts and formulas to use

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Since we have all the info requires we can replace in formula (1) like this:

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