Determine whether the geometric series is convergent or divergent.
10 - 2 + 0.4 - 0.08 + ... If it is convergent, find its sum.

Solution:

Given geometric series 10 - 2 + 0.4 - 0.08 + ...

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

Here the common ratio(r) = -2/10

⇒ r = -0.2,

Which is -1 < -0.2 < 1.

Hence, the given series is convergent.

Sum of series = S_∞ = a/(1 - r)

Here, first term(a) = 10, common ratio(r) = -0.2,

S_∞ = a/(1 - r)

S_∞ = 10/ (1 - (-0.2))

S_∞ = 10/ (1 + 0.2)

S_∞ = 10/1.2

S_∞ = 8.333

Determine whether the geometric series is convergent or divergent.
10 - 2 + 0.4 - 0.08 + ... If it is convergent, find its sum.

Summary:

The given geometric series, 10 - 2 + 0.4 - 0.08 + ..., is convergent and the sum S_∞ is 8.333