Unconditional and unconditional Convergence Coursework

Unconditional and unconditional Convergence - Coursework Example Unconditional and unconditional Convergence: Theorem: Every absolutely convergent series is unconditionally convergent. Conditional Convergence: A convergent series is said to be conditionally convergent if it is not unconditionally convergent. Thus such a series converges in the arrangement given, but either there is some rearrangement that diverges or else there is some rearrangement that has a different sum. Theorem: Every nonabsolutely convergent series is conditionally convergent. In fact, every nonabsolutely convergent series has a divergent rearrangement and can also be rearranged to sum to any preassigned value. The unordered sum of a sequence of real numbers, written as, ∑_iâ‚ ¬Nââ€"’ai has an apparent connection with the ordered sum ∑_(i=1)^∞ââ€"’ai The answer is both have same convergence. Theorem A necessary and sufficient condition for ∑_iâ‚ ¬Nââ€"’ai to converge is that the series ∑_(i=1)^∞ââ€"’ai is absolutely convergent and in this case ∑_(i=1)^∞ââ€"’ã€â€"ai=∑_(i∈â„ µ)ââ€"’aiã€â€"