# Determine whether the series `sum_(1)^oo 0.6^(n-1) - 0.3^n`is convergent or divergent.

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It has to be determined if the series : `sum_(1)^oo 0.6^(n-1) - 0.3^n` is convergent or divergent.

`sum_(1)^oo 0.6^(n-1) - 0.3^n`

=> `sum_(1)^oo 0.6^(n-1) - sum_(1)^oo 0.3^n`

`sum_(1)^oo 0.6^(n - 1)` is a geometric series with first term 1 and common ratio (0.6). The sum of infinite terms of this series is convergent as the common ratio is less than 0 and given by `1/(1-0.6)` = `1/0.4` = 2.5 Similarly `sum_(1)^oo 0.3^n` is equal to `0.3/(1 - 0.3) = 0.3/0.7 = 3/7`

`sum_(1)^oo 0.6^(n-1) - 0.3^n = 2.5 - 3/7 = 29/14`

The series `sum_(1)^oo 0.6^(n-1) - 0.3^n` is convergent

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