adding and removing parentheses in series
We consider series with real or complex terms.
A convergent series can become divergent if one removes an infinite amount of parentheses; cf. the preceding example.
If a series parentheses, they can be removed if the obtained series converges; in this case also the original series converges and both series have the same sum.
If the series
then also the series
converges and has the same sum as (1).
When , we have
by the convergence of (1) to , and
by the condition (2). Therefore the whole partial sum will tend to , Q.E.D.
Note. The parenthesis expressions in (1) need not be “equally long” — it suffices that their lengths are under an finite bound.
|Title||adding and removing parentheses in series|
|Date of creation||2013-03-22 18:54:09|
|Last modified on||2013-03-22 18:54:09|
|Last modified by||pahio (2872)|