In previous lessons, we explored convergence tests that applied to
However, there are series for which the above tests cannot be applied. We could ask: how would we determine if an infinite series converges or diverges if its terms irregularly switch from positive to negative?
Consider for example the recursive sequence
for k = 1, 2, 3, ...
Approximate values of the first eight terms are given in the table below.
k | a_{k} |
---|---|
1 | 1.00000 |
2 | -0.22355 |
3 | +0.11487 |
4 | -0.03180 |
5 | -0.00546 |
6 | -0.00236 |
7 | -0.00076 |
8 | +0.00001 |
We cannot use the integral test (not all terms are positive).
We cannot use the alternating series test (the signs change irregularly).
We will see that we can use the ratio test to show that this series converges.
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