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Limit Comparison Test

LIMIT COMPARISON TEST

Let 09_infinite_series-188.gifand09_infinite_series-189.gif be positive term series and c a positive real number.

Suppose that

aK ≤ cbK       for all infinite K.

Then:

(i) If 09_infinite_series-190.gif converges then 09_infinite_series-191.gif converges.

(ii) If 09_infinite_series-192.gif diverges then 09_infinite_series-193.gif diverges.

PROOF

Assume 09_infinite_series-190.gif converges. Let H and K be infinite. By the Cauchy Convergence Test (Section 9.2).

bH+1 + bH+2 + ... + bK ≈ 0.

Hence

0 ≤ aH+1 + ... + aK ≤ cbH+1 + ... + cbK

= c(bH+1 + ... + bK) ≈ 0.

It follows that

aH+1 + ... + aK ≈ 0

and 09_infinite_series-194.gif converges.

Example 5: Limit Comparison Test

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Last Update: 2006-11-07