Adding endlessly many numbers need not give an endlessly large answer. When each term is a small enough fraction of the last, the running total closes in on a fixed value — and one condition on the ratio decides which happens.
Adding endlessly many numbers sounds like it must give an endlessly large answer. Sometimes it does. But when each term is a small enough fraction of the one before, the running total closes in on a fixed value and stops there — and a single condition on the ratio tells you which of the two will happen.
A divergent series has no such limit — its partial sums grow without bound and no total exists.
An infinite geometric series converges when , and its sum is . When the series diverges and has no sum.
The condition is on the absolute value, so converges just as does. What matters is that each term is smaller in size than the last, no matter which side of zero it sits on.
a)
b)
Deciding takes one division. Everything else about the series is irrelevant until this test is passed.
a)
b)
c)
Part (b) is a reminder to test before computing. Feeding a divergent series into the formula produces a number, but that number is meaningless.
| Property | Convergent | Divergent |
|---|---|---|
| Condition on | ||
| Sum | None | |
| Behaviour of | Settles toward a fixed value | Grows without bound |
| Example |