State why the series the summation from n equals 1 to infinity of the quotient of 1 and 3 raised to the n minus 1 power is convergent or divergent, and if it is convergent, find the sum.

Question

State why the series  is convergent or divergent, and if it is convergent, find the sum.

Answer 1

This is a geometric series with r = 1/3. This infinite series converges absolutely provided -1 < r < 1, to the limit S_∞ = a₁ / (1 - r) = 1 / 1/3 = 3.