Need to know if the Series converges, and its sum

The series given in the problem is `sum_(n=0)^oo (2^n-1)/3^n` .

This is a convergent series.

The nth term of the series is `T_n = (2^n-1)/3^n` .

`sum_(n=0)^oo (2^n-1)/3^n`

= `sum_(n=0)^oo 2^n/3^n-1/3^n`

= `sum_(n=0)^oo (2/3)^n-sum_(n=0)^oo 1/3^n`

= `1/(1 - 2/3) - 1/(1 - 1/3)`

= `1/(1/3) - 1/(2/3)`

= `3 - 3/2`

= `3/2`

The required sum `sum_(n=0)^oo (2^n-1)/3^n = 3/2`

Comments

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