http://www.ck12.org Chapter 8. Infinite Series
8.3 Series Without Negative Terms
Learning Objectives
- Demonstrate an understanding of nondecreasing sequences
- Recognize harmonic series, geometric series, andp−series and determine convergence or divergence
- Apply the Comparison Test, the Integral Test, and the Limit Comparison Test
Nondecreasing Sequences
In order to extend our study on infinite series, we must first take a look at a special type of sequence.
Nondecreasing Sequence
Anondecreasing sequence{Sn}is a sequence of terms that do not decrease:
S 1 ≤S 2 ≤S 3 ≤...≤Sn≤....Each term is greater than or equal to the previous term.
Example 1 5 , 10 , 15 , 20 ,...is a nondecreasing sequence. Each term is greater than the previous term: 5< 10 < 15 <
20 <....
10 , 000 , 1000 , 100 ,...is not a nondecreasing sequence. Each term is less than the previous term: 10, 000 ,> 1000 ,>
100 ....
3 , 3 , 4 , 4 , 5 , 5 ,...is a nondecreasing sequence. Each term is less than or equal to the previous term: 3≤ 3 ≤ 4 ≤ 4 ≤
5 ≤ 5 ≤....
A discussion about sequences would not be complete without talking about limits. It turns out that certain nonde-
creasing sequences are convergent.
Theorem
Let{Sn}be a nondecreasing sequence:S 1 ≤S 2 ≤S 3 ≤...≤Sn≤....
nlim→∞Sn- If there is a constantBsuch thatSn≤Bfor alln, then limn→∞Snexists and limn→∞Sn=LwhereL≤B.
- If the constantBdoes not exist, then limn→∞Sn= +∞.
The theorem says that a bounded, convergent, nondecreasing sequence has a limit that is less than or equal to the
bound. If we cannot find a bound, the sequence diverges.
Example 2Determine if the sequence{ 6 nn+ 5 }converges or diverges. If it converges, find its limit.
Solution
Write the first few terms: 111 , 172 , 233 , 294 ,.... The sequence is nondecreasing. To determine convergence, we see if
we can find a constantBsuch that 6 nn+ 5 ≤B. If we cannot find such a constant, then the sequence diverges.