Sequence & Series

A sequence is a set of terms arranged in a particular order such as

1. (Finite sequence)
2. (Infinite sequence)
3. (Finite sequence)

while a series is the summation of the terms in a sequence, e.g.

1.  (Finite series)
2. (Infinite series)
3. (Finite series)

A sequence is a convergent sequence if and only if there is a finite number a where . Otherwise, the sequence is divergent. Likewise, a series is a convergent series if and only if there is a finite number S where . Otherwise, the series is divergent.

For example, harmonic sequence is convergent but its series is divergent. Perhaps putting the sequence and series in graph might help you to understand better. Look at the graph below, y-coordinate of point A represents the harmonic sequence while y-coordinate of point B represents summation of the harmonic series, where n is the index. Drag the value of n and observe the positions of point A & B. You'll find that point A is approaching to x-axis (y = 0) as n is getting bigger but point B keeps on rising.



Let's take a look at another type of divergent series - Geometric series with common ratio :
.
The points in purple represent sequence and points in red represent series. Tick on the series you want to look into, and drag the values of a and/or n to observe the convergence of the sequence and series.




Now, consider 3 types of infinite convergence series:

1. Geometric series with common ratio, 0 < r < 1 :
   


2. Geometric series with common ratio, -1 < r < 0 :
   


3. Alternating harmonic series :
   

Graph shown below is the graph of Geometric series with and . The point in purple represents the respective sequence and the red represents the series. Look at the graph below, y-coordinate of point A represents the geometric sequence while y-coordinate of point B represents summation of the geometric series. n is the index. Drag the value of n and observe the positions of point A & B. You'll find that point A is approaching to x-axis (y = 0) as n is getting bigger but point B tends to stop rising.

Try to play around with the graph below. After that, click on other choices of a and r to observe the convergence of the sequence and series given.



Betty, Created with GeoGebra