微積分積分鑑定法與p-級數

7.3

The Integral Test and



_{−Series (Positive}

Series)

The Integral Test:

Theorem 85 _{If  is positive, continuous, and decreasing for  ≥ 1 and} =  ()  then ∞ X =1  and Z ∞ 1  ()  either both converge or both diverge.

Example 182 Apply the Integral Test to the series P∞_=1  2₊₁

Example 183 Apply the Integral Test to the series P∞_=121₊₁

p-Series and Harmonic Series: The series ∞ X =1 1  = 1 1 + 1 2 +· · · 

is a p-series, where  is a poistive constant. For  = 1 the series

∞ X =1 1  = 1 1 + 1 2 + 1 3 +· · · is the Harmonic series.

Theorem 86 _{The −series}

∞ X =1 1  = 1 1 + 1 2 +· · ·

() converges if   1 and () diverges if 0  _{≤ 1}

Example 184 () From the above Theorem, it follows that the harmonic series ∞ X =1 1  = 1 1 + 1 2 + 1 3 +· · · 101

diverges.

() From the above Theorem, it follows that the harmonic series

∞ X =1 1 2 = 1 12 + 1 22 + 1 32 +· · · converges.

Example 185 Determine whether the following series converges or diverges

∞

X

=2

1

 ln  (Use the Integral Test)