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It can be hard to determine which convergence test to apply to a given infinite series. While is tempting to search for an optimal order of convergence tests that can be applied to any given infinite series, a more efficient approach is to develop a firm understanding of what requirements each of the tests require to be applied. This way, certain convergence tests can be ruled out or considered by simple inspection of the form of the infinite series.
Infinite Series Forms
Knowing what convergence test to apply for a given series can involve classifying the series according to its form. For example, if the series has the form Σ 1/np, then the series is a p-series, which we know is convergent if p is greater than 1, and divergent otherwise.
| Form | Convergence Test | Conditions for Convergence |
|---|---|---|
| Σ 1/np | p-series | p > 1 |
| Σ ark-1 | geometric series | |r| < 1 |
| Σ (-1)kak | alternating series | ak → 0 as k → ∞ |
| Σ ak, ak = f(k), and f is continuous, positive and decreasing | integral test | integral of f(x) from 1 to infinity exists |
In all convergence tests, some preliminary algebraic manipulation may be required to bring a series into one of these forms. Moreover, most tests can also tell you if a series diverges, and all tests yield inconclusive results if they cannot be applied.
The Ratio Test
The ratio test doesn't fit into the above table as well as the other tests, but can be applied to any infinite series. It is often helpful when the general term of the series contains factorials, or constants raised to a power of k.
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