http://www.ck12.org Chapter 8. Infinite Series
8.4 Series With Odd or Even Negative Terms
Learning Objectives
- Demonstrate an understanding of alternating series
- Apply the Alternating Series Test to an appropriate series
- Explain the difference between absolute and conditional convergence
- Determine absolute and/or conditional convergence of series
Alternating Series (harmonic, geometric, p-series)
Alternating seriesare series whose terms alternate between positive and negative signs. Generally, alternating series
look like one of these expressions:
u 1 −u 2 +u 3 −u 4 +...or−u 1 +u 2 −u 3 +u 4 −....
Either the terms with the even indices can have the negative sign or the terms with the odd indices can have the
negative sign. The actual numbers represented by theu′isare positive.
There are several types of alternating series. One type is thealternating harmonic series:
∞
k∑= 1 (−^1 )k+^11 k=^1 −^12 +^13 −^14 +....This series has terms that look like the harmonic series but the terms with even indices have a negative sign.
Another kind is thealternating geometric series.Here is one example:
∞
k∑= 1 (−^1 )k( 2
3
)k− 1
=− 1 +^23 −( 2
3
) 2
+
( 2
3
) 2
−....
The odd-indexed terms of this series have the negative sign.
Thealternatingp−seriesis another type of alternating series. An example could look like this:
∞
k∑= 1 (−^1 )k−^11
√ (^3) k= 1 −
1
√ 32 +
1
√ 33 −
1
√ 34 −....
From all of these examples, we can see that the alternating signs depend on the expression in the power of−1 in the
infinite series.