00:01
In this question, we are asked to determine if the given series converges or diverges.
00:06
To do that, note that we can rewrite this series as a series 2 to the k over 3 to the k, plus 1 over 3 to the k, k from 0 to infinity.
00:20
And we can split this series into 2 series, the series 2 thirds to the k, plus the series 1 thirds to the k.
00:38
Each of the series is a geometric series, and the common ratio, of the first series, let's call it r1 equals to two -thirds.
00:58
And the first term of the first series is 1, right? because when k equals 0, we are going to get 1.
01:07
And since r1 is less than 1, this means that the series converges.
01:14
And the sum of the series equals to a1 over 1 minus r1 and equals to 1 over 1 minus 2 thirds.
01:27
Equals to 1 over 1 3 equals to 3...