IB Math SL Type I/1 Portfolios IA Task Infinite Summation Help Tutors Examples Samples

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Aim: The aim of this portfolio is to research about the summation of infinite series. In first place I will consider the general sequence with constant values for and variables for . The , which is calculate the sum of the first terms of the above sequence for will be calculated with the programme of Microsoft Excel and illustrated in graphs with Graphmatica; another computer based program, so that we can create a general statement which can be proven. Furthermore, to expand the investigation I will explore the same general sequence, but with variables in both and . I will accomplish this by doing the same method as the first exercises. Finally, I will conclude by showing the scopes/limitations of the general statement. Where, , , , , ... We must take into account that factorial notation is in succession, this means that the factorial notation shows all the natural numbers from 1 to , in such way that: Following the first exercise, we will change the variables of terms and , in such way that and . If we replace these values into the original sequence we obtain: , , , , ... In order to calculate the sum of the first terms of the above sequence for we must know that the sum of the first 10 values is in progression. Hence, I will use Microsoft Excel in order to plot results in a suitable table. The first column will contain the different values of , which come from 1 to 10. The second column will contain the results obtained by replacing each of the values in the form of . And the third column will contain the gradual sum of each of the terms obtained in the second column. For example, the first value of the third column will be added to the second value of the second column giving the second value of the third column, and so on. With the help of Excel the table will look like this: Replacing in the form 0 1 1 1 0,69314718 1,69314718 2 0,24022651 1,93337369 3 0,05550411 1,9888778 4 0,00961813 1,99849593 5 0,00133336 1,99982928 6 0,00015404 1,99998332 7 1,5253E-05 1,99999857 8 1,3215E-06 1,99999989 9 1,0178E-07 1,99999999 10 7,0549E-09 2 Table 1: Shows the sum of the first 10 terms for After analysing the previous results the sum of the first terms of the above sequence for is 2. Likewise, we can represent the relationship between and in a graph by replacing the values of in the y-axis and the values of in the x-axis

Infinite Summation IB Math SL Portfolio - Type I
A. Introduction
Aim: In this task, you will investigate the sum of infinite sequences , where

A notation that you may find helpful in this task is the factorial notation , defined by

e.g. 5 Note that 0

B. Consider the following sequence of terms where and .



I. Calculate the sum of the first n terms of the above sequence for . Give your answers correct to six decimal places.

II. Using technology, plot the relation between and n. Describe what you notice from your plot. What does this suggest about the value of as n approaches ?

C. Consider another sequence of terms where and .



I. Calculate the sum of the first n terms of the above sequence for . Give your answers correct to six decimal places.

II. Using technology, plot the relation between and n. Describe what you notice from your plot. What does this suggest about the value of as n approaches ?

D. Now consider a general sequence where .

I. Calculate the sum of the first n terms of this general sequence for for different values of a. Give your answers correct to six decimal places.

II. Using technology, plot the relation between and n. Describe what you notice from your plot. What does this suggest about the value of as n approaches ?

III. Use your observations from these investigations to find a general statement that represents the infinite sum of this general sequence.

E. Now we will expand our investigation to determine the sum of the infinite sequence , where



Define as the sum of the first n terms, for various values of a and x,

e.g. is the sum of the first nine terms when and .

I. Let . Calculate for various positive values of x.

II. Using technology, plot the relation between and x. Describe what you notice from your plot.

III. Let . Calculate for various positive values of x.

IV. Using technology, plot the relation between and x. Describe what you notice from your plot.

F. Continue with this analysis to find the general statement for as n approaches .

I. Test the validity of the general statement with other values of a and x.

II. Discuss the scope and/or limitations of the general statement.

III. Explain how you arrived at the general statement.

G. Conclusion

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:

International Baccalaureate Math Standard Level Internal Assessment Portfolio Type: I Portfolio Title: Infinite Summation : 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. Now, we need to find the sum of Sn : Now, using Excel 2010, let's plot the relation between Sn and n : Looking at the graph, we can notice that Sn increases rapidly at first, and then it evens out when it reaches 2, which seems like an asymptote. The same happens with the terms' values. They decrease rapidly until they reach the 0, which if we plot will seem like its asymptote. Therefore, we can see that both move a maximum of 1 unit away from their first point, and then even out to the mentioned asymptote. For Sn , the asymptote is . For the terms' calculation for given n, the asymptote is . Therefore: As n approaches infinity, Sn approaches 2: , Now, we do the same thing as before, but for with the same condition for n (): Now we have to calculate Sn again, but for : We plot the relation between Sn and n for this case (using Excel 2010) Looking at the graph, we can see the same happening as in the previous graph, however Sn 's asymptote was

The following expression is an example of an infinite surd. Find the formula for an+1 in terms of a a1 = a2 = a2 = a3 = a3 = an+1 = an = Calculate the decimal values of the first ten terms of the sequence a1 = 1.414213562373100 a2 = 1.553773974030040 a3 = 1.598053182478620 a4 = 1.611847754125250 a5 = 1.616121206508120 a6 = 1.617442798527390 a7 = 1.617851290609670 a8 = 1.617977530934740 a9 = 1.618016542231490 a10 = 1.618028597470230 Using technology, plot the relation between n and an. Describe what you notice. 21 1.618033988736670 22 1.618033988745810 23 1.618033988748630 24 1.618033988749500 25 1.618033988749770 26 1.618033988749860 27 1.618033988749880 28 1.618033988749890 29 1.618033988749890 30 1.618033988749890 31 1.618033988749890 32 1.618033988749890 33 1.618033988749890 34 1.618033988749890 35 1.618033988749890 36 1.618033988749890 37 1.618033988749890 38 1.618033988749890 39 1.618033988749890 40 1.618033988749890 n an 1 1.414213562373100 2 1.553773974030040 3 1.598053182478620 4 1.611847754125250 5 1.616121206508120 6 1.617442798527390 7 1.617851290609670 8 1.617977530934740 9 1.618016542231490 10 1.618028597470230 11 1.618032322752000 12 1.618033473928150 13 1.618033829661220 14 1.618033939588790 15 1.618033973558280 16 1.618033984055430 17 1.618033987299220 18 1.618033988301610 19 1.618033988611370 20 1.618033988707090 By plotting the relation between n and an, one notices that as n increases, an increases. However this increase is not proportional to the increase of n, an seems to be increasing towards 1.62. Once n reaches 28 an ceases to increase, remaining stable at 1.618033988749890. This suggests that as n becomes very large an - an+1 = 0 As such, we can conclude that the exact value for this infinite surd is 1.618033988749890. Consider another infinite surd: Find the formula for an+1 in terms of a a1 = a2 = a2 = a3 = a3 = an+1 = an = Calculate the decimal values of the first ten terms of the sequence a1 = 1.847759065 a2 = 1.9615705608 a3 = 1.9903694533 a4 = 1.9975909124 a5 = 1.9993976374 a6 = 1.9998494037 a7 = 1.9999623506 a8 = 1.9999905876 a9 = 1.9999976469 a10 = 1.9999994117 Using technology, plot the relation between n and an. Describe what you notice. 21 2.0000000000 22 2.0000000000 23 2.0000000000 24 2.0000000000 25 2.0000000000 26 2.0000000000 27 2.0000000000 28 2.0000000000 29 2.0000000000 30 2.0000000000 31 2.0000000000 32 2.0000000000 33 2.0000000000 34 2.0000000000 35 2.0000000000 36 2.0000000000 37 2.0000000000 38 2.0000000000 39 2.0000000000 40 2.0000000000 n an 1 1.847759065 2 1.9615705608 3

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