Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
10 - 8 + 6.4 - 5.12 +…

Solution:

Consider 10 - 8 + 6.4 - 5.12 + ...

A geometric progression will be convergent if the common ratio of the series is between -1 and 1.

Here the common ratio,

r = -8/10

⇒ r = -0.8

Which is -1 < -0.8 < 1.

Hence, the given series is convergent.

Sum of series = S_∞ = a/(1 - r)

Here, first term(a) = 10,

common ratio(r) = -0.8,

S_∞ = a/(1 - r)

⇒ S_∞ = 10/(1 - (-0.8))

⇒ S_∞ = 10/(1 + 0.8)

⇒ S_∞ = 10/1.8

⇒ S_∞ = 5.555

Therefore, the given series is convergent and the sum S_∞ is 5.555

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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
10 - 8 + 6.4 - 5.12 +…

Summary:

The given geometric series, 10 - 8 + 6.4 - 5.12 +..., is convergent and the sum S_∞ is 5.555