Previous: Proof of the Ratio Test
Next: Absolute Convergence Implies Convergence
Previous: Proof of the Ratio Test
Next: Absolute Convergence Implies Convergence
Previous: Proof of the Ratio Test
Next: Absolute Convergence Implies Convergence
The ratio test requires the idea of absolute convergence. Given any infinite series Σa_{k}, we can introduce the corresponding series
whose terms are the absolute values of the original series. We can explore whether this corresponding series converges, leading us to the following definition.
Definition: Absolute Convergence |
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The infinite series
is absolutely convergent if the series of absolute values is convergent. |
A few simple examples demonstrate the concept of absolute convergence.
The infinite series
is absolutely convergent because
is a convergent p-series (p =2).
The infinite series
is convergent (by the alternating series test), but is not absolutely convergent because
is the infamous harmonic series, which is not a convergent series.
It is possible for a series to be convergent, but not absolutely convergent (such series are termed conditionally convergent, but we do not need this definition for our purposes).
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