00:01
In this question, we are asked to determine whether the series converges or diverges, and if it converges we are asked to find its sum.
00:07
We will first rewrite this series as a sum of two series.
00:11
The series 3 to the n over 7 to the n, and the c plus the series 2 to the n over 7 to the n.
00:27
Then we can rewrite them as the series 3 over 7 to the n plus the series 2 over 7 to the n.
00:44
Both series are geometric series.
00:51
The common ratio of the first series is 3 over 7.
01:04
The absolute value of the common ratio, which is still 3 over 7, is less than 1, and that means that the series converges.
01:16
And its sum is a divided by 1 minus r.
01:20
We can calculate its sum only because this is a geometric series.
01:25
A in this formula is the first term of the series, and to get a simply plug in the first value of the index of summation in the general formula, and we will get 3 over 7.
01:38
So the sum equals to 3 over 7 divided by 1 minus 3 over 7.
01:46
That equals to 3 over 7 multiplied by 7 over 4, because 1 minus 3 over 7 is 4 over 7, and this equals to 3 over 4...