Infinite Summation

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  • Topic: Series, Addition, Natural logarithm
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IB Mathematics SL I IA Lab: Infinite Summation

Kristen Rodríguez Mr. Zivkovich May 2010

1

When one considers the concept of infinite summation, there are many observable patterns between the exponent on the numerator and both the result and the sum of the series. This investigation aims to explore this concept of infinite summation and finally arrive at a general statement to determine the sum of the infinite sequence tn. The analysis will also discuss the scope and limitations of the general statement and test the validity by using various values of a and x. Conclusions will be supported by mathematical examples. The following function will be considered, where x and a remain constant throughout the entirety of the sequence and the numerator is raised to the next consecutive number. Tn = 



 

   

As observed, a is the argument of the natural logarithm (ln), x is the value by which the natural logarithm is multiplied and n represents the number of terms in the infinite sequence. To facilitate the exploration of infinite summation, the following example will be worked out: 1)  Ž  Ž   Ž    

When solved, the answers to the above sequence in decimal form are: 1, .6931471806, .240226507, .0555041087 « It can be observed that as the exponent increases, the result tends to zero. Thus, for this sequence it can be deduced that as the n«.. result tends to zero. Sum (Sn) of the first n terms of the above sequence for 0 ” n ” 10   Ž  Ž    Ž Ž           = 1.999999 § 2    Ž      

             

2

To further investigate the relationship between Sn and n, they will be plotted, using technology. Relation between Sn and n (when a=2) n Sn 0 1.000000 1 1.693147 2 1.93373688 3 1.9888778 4 1.99849593 5 1.99982928 6 1.99998338 7 1.99999857 8 1.99999989 9 1.99999999 10 2.000000 A very interesting pattern can be observed from the above plot. As n increases, Sn tends to 2. Thus, as n approaches ’ Sn tends to a. To test the tendency found in the above sequence, the following example will be worked out: 2)  Ž  Ž   Ž    

When solved, the answers to the above sequence (x=1,a=3) in decimal form are: 1, , , « It can be observed that the pattern found in example 1 holds true after n=2. Thus, for this sequence it can be deduced that as the n«.. result tends to zero. Sum (Sn) of the first n terms of the above sequence for 0 ” n ” 10   Ž  Ž    Ž      = Ž      §3  Ž      

               

3

To further investigate the relationship between Sn and n, they will be plotted, using technology. Relation between Sn and n (when a=3) n 0 1 2 3 4 5 6 7 8 9 10 Sn 1 2.098123 2.702087 2.923082 2.983779 2.997115 2.999557 2.99994 2.999993 2.999999 3

The tendency observed in example 1 still holds true (as n approaches ’ Sn tends to a). In this series, as n increases, Sn tends to 3. To investigate whether this pattern holds true for all like sequences where x=1, the following examples will be tested: 3)  Ž  Ž   Ž    

When solved, the answers to the above sequence (x=1,a=4) in decimal form are: 1, , , « It can be observed that the pattern found in example 2 holds true (after n=2 the result tends to zero). Sum (Sn) of the first n terms of the above sequence for 0 ” n ” 10     Ž  Ž    Ž      = Ž      §4  Ž      

               

4

To further investigate the relationship between Sn and n, they will be plotted, using technology. Relation between Sn and n (when a=4) n 0 1 2 3 4 5 6 7 8 9 10 Sn 1 2.386294 3.3472 3.791233 3.945123 3.987791 3.997649 3.999601 3.99994 3.999992 3.999999

The tendency observed in example 1 still holds true (as n approaches ’ Sn tends to a). In this series, as n increases, Sn tends to 4. 4)  Ž  Ž   Ž    

When solved, the answers to the above sequence(x=1, a=5) in decimal form are: 1,...
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