7.4
Comparisons of Series (Positive Series)
Direct Comparison Test:
In the following pairs, the second series cannot be tested by the same convergence test as the first even thougn it is similar to the first.
1. P∞=0 21 is geometric, but P∞ =0 2 is not. 2. P∞=0 13 is a −series, but P∞ =0 1 3+1 is not.
3. = (2+3) 2 is easily integrated, but =
2
(2+3)2 is not.
In this section you will study two additional tests for positive -term series.
(1) Dirct Comparison Test. (2) Limit Comparison Test.
Theorem 87 (Dirct Comparison Test) Let 0 ≤ for all greater
then some integer
1. If P∞=1 converges, then
P∞
=1 converges. (If the ”larger” series
converges, the ”smaller” series must also converge.) 2. If P∞=1 diverges, then
P∞
=1 diverges. (If the ”smaller” series
diverges, the ”larger” series must also diverges.)
Example 186 Determine the convergence or divergence of IfP∞=1 2+31
1 2 + 3
1
3 ≥ 1
Example 187 Determine the convergence or divergence of IfP∞=1 2+1√ 1 2 +√ ≤ 1 √ for ≥ 1 and 1 ≤ 1 2 +√ for ≥ 4 103
Theorem 88 (Limit Compareson Test) Suppose that 0 0 and lim →∞ µ ¶ =
where is finite and positive. Then the two series P and P either
both converges or both diverges.
Example 188 Show that the following series diverges
∞ X =1 1 + 0 0 Use ∞ X =1 1
Technique: Cancel to lower order term and Comparison −series. Example 189 () ∞ X =1 1 32− 4 + 5 ∞ X =1 1 2 () ∞ X =1 1 √ 3− 2 ∞ X =1 1 √ () ∞ X =1 2 − 10 45+ 3 ∞ X =1 1 3
Example 190 Determine the convergence or divergence of
∞ X =1 √ 2+ 1
Example 191 Determine the convergence or divergence of
∞ X =1 2 43+ 1 104
