Post on 13-Dec-2015
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!
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,
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
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.
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.
Example:
series diverges
Converges to a/(1-r)if |r|<1. Diverges if
|r|>1
nth-Term Test Is lim an=0 no
GeometricSeries Test Is Σan = a+ar+ar2+ … ?
yes