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SL IA Type 1: Infinite Summation Portfolio

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Introduction

Example Math Portfolio SL

Infinite Summation – Type 1:

We being this analysis by investigating series of the type , where tn represents the nth term in the series, x and a are two parameters that may be varied and n! = n X (n -1) X (n -2) X … X 3 X 2 X 1.

Initially we will consider the summation, Sn where Sn means to sum the first n terms of the series.  We will find the Sn for the

Middle

1.999999

8

2

9

2

10

2

To further clarify how these values were obtained a sample calculation is shown below: , a screenshot highlighting the formula used in Excel to complete the rest is shown in figure 1 below. Figure 1: Showing how Excel was used to calculate Sn

If we now take these values and create a graph of n vs. Sn we obtain the result shown in Figure 2

From this graph it is clearly seen that the summation Sn converges on the value 2.  It can be said then that as n approaches infinity the summation converges on the value 2. If we complete this same exercise, but this time change the value of parameter a to equal 3 we obtain the following results for Sn shown as a screenshot for Excel. Conclusion

So clearly as the value of a decreases the value of Sn does converge on it, however it does not do so in a smooth systematic way, but rather chaotically.

Now we will try large values of a:

a = 1000 Now it does not converge to the value of a, after 10 steps; however if the value of n is increased the value of Sn does indeed converge on 1000 as shown.  It also appears to do so in a smooth fashion. We can clearly observe a general trend for Sn in which assuming n is large enough (i.e n approaches infinity) then the value of Sn converges upon the value of a.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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