Sequence and Series
- Author:
- Dr. Jack L. Jackson II
Sequence and Series
A sequence is a function whose domain is the set of natural numbers. A series is the sum of the terms of a sequence.
In the app: enter appropriate functions for the sequence {a_k} using k as the independent variable. The graph will be displayed as a sequence of isolated blue dots. We can compare two series by entering a second formula for the sequence {b_k}.
For each sequence, if we sum the first k terms, then we obtain a sequence of partial sums. These are displayed when we check the checkbox for partial sums.
In the spreadsheet view we see the numerical values for the terms of the sequences and the sequence of partial sums. Both the spreadsheet and the graph display the first 100 terms.
If the limit of the sequence of partial sums as the number of terms approaches infinity exists, then we say the infinite series converges to this limit, and we take this limit as the value of the infinite sum. However, the limit of partial sums may not exist, in which case we say the infinite series diverges.
On of the things that we are often concerned about is if an infinite series converges or diverges. There are several tests to determine if the series converges or diverges. Some of them can be illustrated with this activity.
The first is the Divergence Test, which says that a series will diverge if the limit of the underlying sequence is not zero. Looking at the values of the sequence in the spreadsheet and the graph of the points on the sequence, we may be able to see when this limit is not zero, and thus the series diverges by the divergence test.
Another test is the Direct Comparison Test. In this test we compare the series of concern, , to a known series . We can use the graphs and spreadsheet to illustrate this comparison.
In the Limit Comparison Test, we look at the ratio of the two . The first 100 terms of this ratio are given in the spreadsheet.
In the Ratio Test, we look at the ratio between successive terms from the original sequence, . These values are also given in the spreadsheet.