Determine whether the geometric series is convergent or divergent. 
10 - 4 + 1.6 - 0.64 +... If it convergent, find the sum?

Solution:

Given, the geometric series is:

10 - 4 + 1.6 - 0.64 +...

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

Here the common ratio,

r = -4/10

⇒ r = -0.4,

Which is -1 < -0.4 < 1.

Hence, the given series is convergent.

Sum of series = S_∞ = a/(1 - r)

Here, a (first term) = 10, r (common ratio) = -0.4

S_∞ = a/(1 - r)

⇒ S_∞ = 10/(1 - (-0.4))

⇒ S_∞ = 10/(1 + 0.4)

⇒ S_∞ = 10/1.4

⇒ S_∞ = 7.14

Determine whether the geometric series is convergent or divergent.
10 - 4 + 1.6 - 0.64 +... If it convergent, find the sum?

Summary:

The given geometric series, 10 - 4 + 1.6 - 0.64 +..., is convergent and the sum S_∞ is 7.14