Weare given the series:
2/1 + 4/3 + 8/9 + 16/27 + ...
and are asked to find the infinite sum of it.
so we notice that we are in the presence of a geometric sequence with common ration given by: 2/3
Which we ontained by making the ratios of one term and the preceeding one:
8/9 divided by 4/3 = 2/3, and this is common to all the elements of the series.
So, we use the formula for the infinite sum of a geometric sequence (which is true as long as the common ratio is samller than 1 - true in our case):
Infinite sum = a1 / (1 - r)
where a1 is the first term of the sequence, and "r" the common ratio.
So, in our case se have:
Infinite sum = a1 / (1 - r) = 2 / (1 - 2/3) = 2 / (1/3) = 2 * 3 = 6
Then , the infinite sum of this series is "6".
: