manipulating convergent series

The of the series in the following theorems are supposed to be either real or complex numbersMathworldPlanetmathPlanetmath.

Theorem 1.

If the series  a1+a2+  and  b1+b2+converge and have the sums a and b, respectively, then also the series

(a1+b1)+(a2+b2)+ (1)

converges and has the sum a+b.

Proof.  The nth partial sum of (1) has the limit

Theorem 2.

If the series  a1+a2+  converges having the sum a and if c is any , then also the series

ca1+ca2+ (2)

converges and has the sum ca.

Proof.  The nth partial sum of (2) has the limit

Theorem 3.

If the of any converging series

a1+a2+a3+ (3)

are grouped arbitrarily without changing their , then the resulting series

(a1++am1)+(am1+1++am2)+(am2+1++am3)+ (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