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Home Infinite Series Properties of Infinite Series Corollaries: Convergent and Divergent Sums
 
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Corollaries: Convergent and Divergent Sums

COROLLARY 1

If 09_infinite_series-138.gif converges, then the tails 09_infinite_series-139.gif approach zero as m approaches x,

09_infinite_series-140.gif

PROOF If H is infinite, then

09_infinite_series-141.gif

so09_infinite_series-142.gif

COROLLARY 2

If a series 09_infinite_series-143.gif converges, then it remains convergent if finitely many terms

are added, deleted, or changed.

PROOF

If am is the last term changed, then the tail

09_infinite_series-144.gif

is left unchanged, so it still converges.

Warning: Although the convergence properties of a series are not affected by changing finitely many terms, the value of the sum, if finite, is affected.
Home Infinite Series Properties of Infinite Series Corollaries: Convergent and Divergent Sums

Last Update: 2006-11-07