Examine the sequence 1 + 1/3 + (1/3)^2 + … + (1/3)^n Under what condition does it converge? What does it converge to?

This is a sequence in the form 1 + c + c² + … cⁿ

1 + c + c² + … cⁿ  * (1 - c) = 1 – cⁿ⁺¹

So we can rewrite the sequence as

1 – cⁿ⁺¹ / 1 - c

Which is the same as

1 / 1 – c   –  cⁿ⁺¹ / 1 - c

Since 0 <= c < 1, We can see that the second term approaches 0 as n approaches infinity

Hence as n approaches infinity the sequence converges to 1 / 1 - c

Substitute c for 1 / 3 and we get 1 / (1 – (1/3)) = 1 / (2/3) = 3 / 2