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,

¿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.

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

Cauchy’s Condensation Test

be a decreasing sequence of positive

and are either convergent or divergent.

End of chapter 1