Unconditional and unconditional Convergence - Coursework Example

Handout Section 3.7 and 3.8 Any finite sum may be rearranged and summed in any order because addition is commutative but it isn’t the case with series. Unconditional and unconditional Convergence: A series is said to be unconditionally convergent if all rearrangements of that series converge and have the same sum. Those series that do not allow this but do converge are called conditionally convergent. 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, has an apparent connection with the ordered sum The answer is both have same convergence.

Theorem A necessary and sufficient condition for to converge is that the series is absolutely convergent and in this case Products of Series: The rule for the sum of two convergent series = + this is very elementary to prove by applying limits rules. NOTE: The Series is called the formal product of the two series and provided that Products of Absolutely Convergent Series: Theorem: Suppose that is the formal product of two absolutely convergent series and Then converges absolutely too and = * Products of Nonabsolutely Convergent Series: We know that the alternating series is convergent, but not absolutely convergent since the related absolute series is a p-harmonic series with p=1/2.

Theorem: Suppose that is the formal product of two nonabsolutely convergent series and and suppose that this product is known to converge. Then = *