Infinite summation portfolio. A series is a sum of terms of a sequence. A finite series, has its first and the last term defined, and the infinite series, or in other words infinite summation
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Introduction
International Baccalaureate Math Standard Level Internal Assessment Portfolio Type: I Portfolio Title: Infinite Summation Due date: 9th of December, 2011 Teacher: Mr. Peter Vassilev School: The American College of Sofia Candidate Name: Rami Meziad Candidate number: 002368-008 Examination Session: May 2012 Introduction: A series is a sum of terms of a sequence. A finite series, has its first and the last term defined, and the infinite series, or in other words infinite summation [3] is a series which continues indefinitely. The Taylor's theorem [1] and the Euler-Maclaurin's formula [2] will help us solve our given infinite summation, which is: ,,,, And by adding different values for x and a, we will be able to find a general pattern in which the sequences tends to move with. And this is mainly what this portfolio will ask us to do. Method: For our sequence, which is: , we have to substitute in the case where x = 1 and a = 2. After that, we have to calculate the first n terms which happen to be eleven to fulfill the given condition. So after substitution we get Now let's calculate for n, when : In fact, t9 and t10 are not equal to 0, but since we have to take our answers correct to six decimal places, we can't see the real values. However, the numbers become so small, that they become insignificant, or in other words they are equal to 0. ...read more.
Middle
we get: We see that as , We see that and the asymptote (which is also approximately the value of S10) of Sn are the same (in this case the asymptote is 7). It seems that , however we will try with our last value, and see if this is true: For : Now we find S10 for Now, we draw for the last time the relation between Sn and n for (with Excel 2010 again): Sn We note that as , As we were observing all of those different values for a, we've noticed that as , , and also , and if we make an approximate estimation of the Sn, we could easily say that . And this is the general statement that applies in our case. However, theoretically, this only holds true when . We will examine for different values of x, and then tell what is the true general statement for any values for x and a. For now, it is . Now, in order to get a general statement for we will have to change the values for a and x, and then find a pattern. We will start with , and for x we will try various positive values. To compare right the results we'll look correct to 6 decimal places again: : T8 It's obvious that the graph keeps the same way as the one described in all previous graphs. ...read more.
Conclusion
As we have tried decimals for x, but we haven't tried decimals for a we shall do it: For T8 The values of summation reach the asymptote slower than the previous cases but still it is not a limitation of the common behaviour and statement. The same is mentioned when a = ?. For T8 For , the statement is still the same. The last check we can do is with a negative value of x, as for a we cannot do that because of the limitation if the ln function. For T8 Clearly it is the first graph that is distinguishable and doesn't follow the common behavior as well as these are the first values that don't follow the statement Sn = an. The values as well as the graph seem to oscillate under and above the 0 with increasing amplitude of oscillation as much as n approaches infinity. As a conclusion we can say that the general statement Sn = an fulfils quite well the presented infinite summation, including the first case when we stated that Sn = a, which is a particular case when x = 1, so it is according to a1 = a. The statement should consider the following limitations: a = 0, x = 0 - as this will give us a single value "0" and the graph will be a straight line on the x axis. a > 0, x > 0 - as a > 0 is the natural limitation of the ln, and x > 0 shows an oscillating graph. ...read more.
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