## 4.1. Series and Convergence

### Theorem 4.1.6: Absolute Convergence and Rearrangement

Let
be an absolutely convergent series. Then any rearrangement of
terms in that series results in a new series that is also
absolutely convergent to the same limit.

Let
be a conditionally convergent series. Then, for any real
number *c* there is a rearrangement of the series such
that the new resulting series will converge to *c*.

### Proof:

Suppose is absolutely convergent. Then the sequenceconverges. In particular, it is bounded, i.e.S_{n}= |a_{1}| + |a_{2}| + ... + | a_{n}|

*|S*for some number

_{n}| < K*K*. If we take any rearrangement of terms in the series and form a new sequence of partial sums:

thenT_{n}= | | + | | + ... | |

*T*is again bounded by the same number

_{n}*K*. But since all terms in the partial sum

*T*are positive the sequence is monotone increasing. Therefore

_{n}*{T*is monotone increasing and bounded and must therefore converge.

_{n}}It remains to show that the limit of the rearrangement is the same as the limit of the original series. That is left as an exercise.

Finally suppose the series converges conditionally. Let's first collect a few facts:

- By the divergence test (which we will prove later) we know that the sequence of
general terms
*a*converges to zero._{n} - The series does not converge absolutely.
- Since the sequence of partial sums of absolute values is increasing it means that the series of absolute values must "converge" to positive infinity.
- There must be infinitely many positive terms among the
*a*. If we call them_{j}*b*and collect them to form a series then that new series must "converge" to positive infinity (why?)._{j} - There must be infinitely many negative terms among the
*a*. If we call them_{j}*c*and collect them to form a series then that new series must "converge" to negative infinity (why?)._{j}

- Collect enough
*b*so that they add up to a number just bigger than_{j}*c* - Add as many terms from
*c*as is necessary to make the resulting sum just less than_{j}*c* - Add again terms from
*b*to be just bigger than_{j}*c*, then again terms from*c*to be less than_{j}*c*, and so on.

Then you can show that the resulting arrangment of *b _{j}*'s and

*c*indeed forms a series that converges to

_{j}*c*. As usual, the details of this proof are left as an exercise.