Discrete Mathematics Demystified

CHAPTER 13 Series 283
13.11.1 A NEW CONVERGENCE TEST
Suppose that
13.11 The Comparison Test
∞
j=1
is a convergent series of nonnegative terms:
a j
∞
a j = ℓ
j=1
Because the partial sums form an increasing sequence, these partial sums must
increase to ℓ. Therefore for each partial sum SN we may say that
If
N
a j = SN ≤ ℓ
j=1
∞
c j
j=1
is another series satisfying 0 ≤ c j ≤ a j for every j, then the partial sums TN for
this series satisfy
TN =
N
c j ≤
j=1
N
a j ≤ ℓ
Thus the partial sums TN of the series ∞
j=1 c j are increasing (since the c j’s are nonnegative)
and bounded above by ℓ. By a property of bounded increasing sequences
that we have discussed before, we conclude that the sequence of TN ’s converges.
We summarize:
Theorem 13.5 (The comparison test for convergence) Let 0 ≤ c j ≤ a j for every
j. If the series ∞ j=1 a j converges then the series ∞ j=1 c j also converges.
j=1