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.
Again if we create a graph of n versus Sn we obtain the results shown in the figure below.
Again we notice that as n approaches infinity the value of Sn converges to 3 which is equal to the parameter a.
We can begin to notice a pattern that when x = 1 the value of Sn converges to the value of a, as n approaches infinity. In order to verify our assumption we will look at some different values of a.
a = 0.78:
Again we see that as n approaches infinity the series converges to the value of a. It is also interesting to notice that as the value of a decreases the value of Sn seems to converge quicker. To test this hypothesis we will use a = 0.40.
Here it again converges to the value of a, however it does not directly go there instead it oscillates above and below it.
a = 0.002
Here it seems to not converge at all but rather oscillates widely back and forth from positive to negative values. However, if we include a higher value of n (i.e. n = 30) the value of Sn does indeed converge to a.
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.
