8.3. Series Without Negative Terms http://www.ck12.org
If two fractions have the same numerator but different denominators, the fraction with the smaller denominator is
the larger fraction. Thus, 6 nn+ 5 ≤ 6 nn= 61. Then 6 nn+ 5 ≤^16 and, in fact, limn→∞ 6 nn+ 5 =^16.
Series Without Negative Terms (harmonic, geometric, p-series)
There are several special kinds of series with nonnegative terms, i.e., terms that are either positive or zero. We will
study the convergence of such series by studying their corresponding sequences of partial sums.
Let’s start with theharmonic series:
∞
k∑= 11
K=^1 +
1
2 +
1
3 +
1
4 +....
The sequence of partial sums look like this:
S 1 = 1
S 2 = 1 +^12
S 3 = 1 +^12 +^13
S 4 = 1 +^12 +^13 +^14
...
In order for the harmonic series to converge, the sequence of partial sums must converge. The sequence of partial
sums of the harmonic series is a nondecreasing sequence. By the previous theorem, if we find a bound on the
sequence of partial sums, we can show that the sequence of partial sums converges and, consequently, that the
harmonic series converges.
It turns out that the sequence of partial sums cannot be made less than a set constantB. We will omit the proof here,
but the main idea is to show that the selected infinite subset of terms of the sequence of partial sums are greater than
a sequence that diverges, which implies that the sequence of partial sums diverge. Hence, the harmonic series is not
convergent.
We can also work withgeometric serieswhose terms are all non-negative.
Example 3The geometric series∑∞k= 1 (^32 )k−^1 has all non-negative terms. The sequence of partial sums looks like
this: