Conditional and Absolute Convergence
For any sequence, for all n. Therefore,
This means that if converges, then also converges (but not vice-versa).
If the series converges, then the series is absolutely convergent. An absolutely convergent sequence is one in which the length of the line created by joining together all of the increments to the partial sum is finitely long. The power series of the exponential function is absolutely convergent everywhere.
If the series converges but the series diverges, then the series is conditionally convergent. The path formed by connecting the partial sums of a conditionally convergent series is infinitely long. The power series of the logarithm is conditionally convergent.
The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
Read more about this topic: Convergent Series
Famous quotes containing the words conditional and/or absolute:
“Computer mediation seems to bathe action in a more conditional light: perhaps it happened; perhaps it didn’t. Without the layered richness of direct sensory engagement, the symbolic medium seems thin, flat, and fragile.”
—Shoshana Zuboff (b. 1951)
“I tell you, hopeless grief is passionless;
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Beat upward to God’s throne in loud access
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In souls as countries, lieth silent-bare
Under the blanching, vertical eye-glare
Of the absolute Heavens.”
—Elizabeth Barrett Browning (1806–1861)