8.3. SERIES OF REAL NUMBERS
485
1
k 2 +k
to converge or diverge? Why?
(b) The example in the previous part of this exercise illustrates a more general
result. Explain why the Direct Comparison Test, stated here, works.
The Direct Comparison Test. If
0 ≤ b k ≤ a k
for every k, then we must have
0 ≤
b k ≤
a k
1. If
a k converges, then
b k converges.
2. If
b k diverges, then
a k diverges.
Important Note: This comparison test applies only to series with nonnegative
terms.
(i) Use the Direct Comparison Test to determine the convergence or divergence of the series
1
k−1 . Hint: Compare to the harmonic series.
(ii) Use the Direct Comparison Test to determine the convergence or divergence of the series
k
k 3 +1
.