manipulating convergent series
The of the series in the following theorems are supposed to be either real or complex numbers.
Theorem 1.
If the series and converge and have the sums and , respectively, then also the series
(1) |
converges and has the sum .
Proof. The partial sum of (1) has the limit
Theorem 2.
If the series converges having the sum and if is any , then also the series
(2) |
converges and has the sum .
Proof. The partial sum of (2) has the limit
Theorem 3.
If the of any converging series
(3) |
are grouped arbitrarily without changing their , then the resulting series
(4) |
also converges and its sum equals to the sum of (3).
Proof. Since all the partial sums of (4) are simultaneously partial sums of (3), they have as limit the sum of the series (3).
Title | manipulating convergent series |
---|---|
Canonical name | ManipulatingConvergentSeries |
Date of creation | 2013-03-22 14:50:49 |
Last modified on | 2013-03-22 14:50:49 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 12 |
Author | pahio (2872) |
Entry type | Theorem |
Classification | msc 40A05 |
Classification | msc 26A06 |
Related topic | SumOfSeries |
Related topic | MultiplicationOfSeries |