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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 image00.png, 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

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Middle

1.999999

8

2

9

2

10

2

To further clarify how these values were obtained a sample calculation is shown below:

image04.png, a screenshot highlighting the formula used in Excel to complete the rest is shown in figure 1 below.

image05.png

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.image06.png

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.

image07.png

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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

image02.png

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.

image03.png

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.

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