Elementary Calculus: Theorem 2: Tail Of a Series

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Theorem 2: Tail Of a Series

The next theorem corresponds to the Addition Property for integrals,

DEFINITION

The series

is defined as

where

b_n = a_m+n.

This series is called a tail of the original series

THEOREM 2

A series

converges if and only if its tail

converges for any m.

The sum of a convergent series is equal to the mth partial sum plus the remaining tail,

a₁+ ... + a_n + ... = (a₁ + ... + a_m) + (a_m+1 + ... + a_m+n + ...).

PROOF

First assume the tail converges. For any infinite H, we have

a₁ + ... + a_H = (a₁ + ... + a_m) + (a_m+1 + ... + a_H),

Taking standard parts,

Therefore the series converges and

If we assume the series converges we can prove the tail converges in a similar way.

Example 2: Tail of a Geometric Series

Last Update: 2006-Nov-07