## Wednesday, January 11, 2012

### Sequence & Series

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

1. $1,2,3,4,5$ (Finite sequence)
2. $2,4,6,8,10,...$ (Infinite sequence)
3. $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},...,\frac{1}{256}$ (Finite sequence)
while a series is the summation of the terms in a sequence, e.g.

A sequence $\left \{ a_n \right \}$ is a convergence sequence if and only if there is a finite number a where $\lim_{n \to \infty}a_n = a$. Otherwise, the sequence is divergent. Likewise, a series $S_n$ is a convergence series if and only if there is a finite number S where $\lim_{n \to \infty }S_n = S$. Otherwise, the series is divergent.

For example, harmonic sequence is convergence 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. 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 raising.

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Let's take a look at another type of divergence series - Geometric series with common ratio |r| = |a| > 1 :
$a+a^2+a^3+a^4+...,\left | a \right |> 1$.

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.

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Now, consider 3 types of infinite convergence series:
1. Geometric series with common ratio, 0 < r < 1 : $\frac{1}{a}+\frac{1}{a^2}+\frac{1}{a^3}+\frac{1}{a^4}+..., a>1$
2. Geometric series with common ratio, -1 < r < 0 : $\frac{1}{a}+\frac{1}{a^2}+\frac{1}{a^3}+\frac{1}{a^4}+..., a<-1$
3. Alternating harmonic series : $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-...$

Graph shown below is the graph of Geometric series with 0 < r < 1. The point in purple represents the respective sequence and the red represents the series. The solid line in green gives the value of the series as n is getting bigger, whereas the dotted green line tells us the value of the current sum of the series. Try to play around with the graph below. After that, click on other choices to observe the convergence of the sequence and series given.

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Betty, Created with GeoGebra