1.5 Rearrangement of Series Since addition is commutative, any finite sum may be rearranged and summed in any order. If the terms of an infinite series are rearranged into a different order do we get the same result? Answer: No = 1-1+1-1+1-1… = (1 − 1) + (1 − 1) + (1 − 1) + (1 − 1) + ... = 0 + 0 + 0 + 0 + 0 + ... = 0 = 1-1+1-1+1-1… =1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... = 1 + 0 + 0 + 0 + 0 + ... = 1? Something Wrong! Something Wrong !
19
Embed
1.5 Rearrangement of Series Since addition is commutative, any finite sum may be rearranged and summed in any order. If the terms of an infinite series.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1.5 Rearrangement of Series
Since addition is commutative, any finite sum may be rearranged and summed in any order.
If the terms of an infinite series are rearranged into a different order do we get the same result? Answer: No
Be careful! Some operations customary for finite sums might be illegal for infinite convergent sums.
The most famous example is:
∑𝑛=1
∞ (−1 )𝑛−1
𝑛=1− 1
2+ 13−14+ 15−…
is convergent (actually non-absolutely convergent)
So, where
------(i)
Now, If we rearrange this so that every positive term is followed by two negative terms, thus,
¿𝑙2
Grouping these and adding, we obtain
Inserting zeros between the terms of this series, we have
---(i)
---(iii)
----(ii)
(i) and (iii) we get, ----(iV)
(iV) (i)
(The Rearrangement Theorem for Absolutely Convergent Series):
Suppose that converges absolutely,
i.e. converges as well, and
is any arrangement of the sequence {}, then
converges absolutely, and
Dirichlet’s Rearrangement Theorem
Example:
Here’s a similar example:
(The Rearrangement Theorem for conditionally Convergent Series):
Riemann’s Rearrangement Theorem
The use of brackets in an infinite series
THEOREM: If the terms of a convergent series are grouped in parentheses in any manner to form new terms ( the order of the terms remaining unaltered), then the resulting series will converge and converges to the same sum.
i.e. if the series converges to , then the series
also converges to
Proof: Let .
Then, .
, where .
Example1:Consider the series =.Since is convergent, we have i.e. .
Note that . So, by comparison test is convergent.
Example2:
Consider the series
What can you say about the convergence following series where the brackets are removed?
1.6 Other TestsCauchy’s Condensation Test:
Let be a decreasing sequence of positive terms. Then the two series and are either convergent or divergent.