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Theorem 2: Tail Of a Series

The next theorem corresponds to the Addition Property for integrals,

09_infinite_series-124.gif

DEFINITION

The series

09_infinite_series-125.gif

is defined as

09_infinite_series-126.gif

where

bn = am+n.

This series is called a tail of the original series

09_infinite_series-127.gif

THEOREM 2

A series

09_infinite_series-128.gif

converges if and only if its tail

09_infinite_series-129.gif

converges for any m.

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

09_infinite_series-130.gif

or

a1 + ... + an + ... = (a1 + ... + am) + (am+1 + ... + am+n + ...).

PROOF

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

a1 + ... + aH = (a1 + ... + am) + (am+1 + ... + aH),

or 09_infinite_series-131.gif

Taking standard parts,

09_infinite_series-132.gif

Therefore the series converges and

09_infinite_series-133.gif

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: 2006-11-07