How do you find the sum of the infinite geometric series 1 + 2 + 4 + 8 +… ?
2 Answers
This series diverges. Its sum is
Explanation:
A infinite geometric series of the form
In this case the common ratio
The series diverges, that is, it has no finite sum.
Explanation:
For an infinite series to converge to a specific sum, the terms of the series need to approach 0. In this case, not only do the terms not approach 0, but they increase by a factor of 2. It should be intuitive, then, that the series increases without bound, as you are adding an infinite number of increasing values.
.
.
.
For a slightly more rigorous proof that does not rely on knowledge of series, suppose that there exists a number
Let
and let
Note that for any integer
Then we have
But