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Infinite Summation - In this portfolio, I will determine the general sequence tn with different values of variables to find the formula to count the sum of the infinite sequence.

Extracts from this document...

Introduction

Zespół Szkół

Im. Maharadzy Jam Saheba Digvijay Sinhji

„Bednarska”

IB WORLD SCHOOL 1531

image00.png

PORTFOLIO ASSIGNMENT TYPE 1

MATHEMATICAL INVESTIGATION

Set date: Monday, March 8th, 2010

Due date: Monday, March 15th, 2010

                         Name: Tra My Nguyen

                                               Candidate number:

  • Technology used:
  • Microsoft word
  • Rapid-pi
  • Microsoft Excel
  • GDC (TI-84 Plus)

In this portfolio, I will determine the general sequence tn with different values of variables to find the formula to count the sum of the infinite sequence.

I  will investigate the sum of infinite sequences tn, where:

t0 = 1, t1 =  image01.png , t2 =  image06.png, t3 =  image07.png , … , tn =  image08.png , …

        * a must be positive because according to the definition of logarithms:  

y = logxy  image04.png x>0, x image114.png 1,y>0

lna = logea  image04.png a>0

(approximately e = 2.718281828)

* I will use the factorial notation:image128.png

To see how the sum changes when a changes, firstly, I am going to consider the sequence above, where  image94.png  and  image02.png :

image11.pngimage22.png

Let’s define Sn to be the sum of the first (n+1) terms of the sequence,  image33.png .

Using GDC, I will calculate the sums S0, S1, S2, …, S10 (giving answers correct to 6 decimal places):

 S0 = t0 = 1

S1 = S0 + t1 = 1 +  image39.png = 1.693147

S2 = S1 + t2 = 1.693147 +  image43.png = 1.933373

S3 = S2 + t3 = 1.933373 +  image47.png = 1.988877

S4 = S3 + t4 = 1.988877 +  image52.png = 1.998495

S5 = S4 + t5 = 1.998495 +  image63.png = 1.999828

S6 = S5 + t6 = 1.999828 +  image74.png = 1.999982

S7 = S6 + t7 = 1.999982 +  image85.png = 1.999997

S8 = S7 + t8 = 1.999997 +  image89.png = 1.999998322

S9 = S8 + t9 = 1.999998322 +  image90.png = 1.999998424

S10 = S9 + t10 = 1.999998424 +  image91.png = 1.999998431

Now, using Microsoft Excel, I will plot the relation between Sn and n:

n

Sn

0

1

1

1.693147

2

1.933373

3

1.988877

4

1.998495

5

1.999828

6

1.999982

7

1.999997

8

1.999998

9

1.999998

10

1.999998

image92.png

From this plot, I see that the values of Sn increase as values of n increase, but don’t exceed 2, so the greatest value that Sn can have is 2. Therefore, it suggests about the values of Sn to be in domain  image93.png Snimage05.png  2  as n

...read more.

Middle

image124.png = 2.144730

S2 = S1 + t2 = 2.144730 +  image125.png = 2.799933

S3 = S2 + t3 = 2.799933 +  image126.png = 3.049943

S4 = S3 + t4 = 3.049943 +  image127.png  = 3.121492

S5 = S4 + t5 = 3.121492 +   = 3.137873

S6 = S5 + t6 = 3.137873 +  image129.png = 3.140998

S7 = S6 + t7 = 3.140998 +  image130.png = 3.141509

S8 = S7 + t8 = 3.141509 +  image131.png = 3.141582

S9 = S8 + t9 = 3.141582 +  image132.png = 3.141591

S10 = S9 + t10 = 3.131591 +  image133.png = 3.141592

Then, using Microsoft Excel, I plot the relation between Snand n:

n

Sn

0

1

1

2.144730

2

2.799933

3

3.049943

4

3.121492

5

3.137873

6

3.140998

7

3.141509

8

3.141582

9

3.141591

10

3.141592

image134.png

Knowing that  image135.png  (correct to six decimal places), I noticed that in the sequence given where image94.png and  image109.png, Sn increases as n increases, and doesn’t exceed  image36.png . So domain for the infinite sum Snhere is again suggested to be  1 image05.png Snimage05.pngimage36.png .

Now let’s analyse the initial general sequence:

t0 = 1, t1 =  image01.png , t2 =  image06.png, t3 =  image07.png , … , tn =  image08.png , ….

If  I substitute (xlna) with m, I can have a sequence like this:

t0 = 1, t1 = m, t2 =  image136.png , t3 =  image137.png , …

And the sum of these infinite terms is:

image138.png

On the other hand, as defined by power series expansion, we have:

image03.png

Therefore, we see that the infinite sum can be counted by em, where m = xlna.

So what I noticed from here is (hypothesis):

  • Values of Sn  increase as values of  n increase.
  • The greatest value for Sn infinite is suggested to be a.   image04.png  1 image05.png Snimage05.png  a.
  • The statement to find the sum of infinite sequence is suggested to be e(xlna).

Now, it would be very interesting to expand this investigation to determine the sum of the infinite sequence tn, where:          

t0 = 1, t1 =  image01.png , t2 =  image06.png, t3 =  image07.png , … , tn =  image08.png , …

Tn (a,x) is defined to be the sum of the first n terms, for variable values of a and x.

E.g.: T6 (2,3) is the sum of the first 6 terms when a = 2 and x = 3.

Let a = 2. I will  calculate T9 (2,x)

...read more.

Conclusion

T9 (2,x) and x.

When a = 2 and x = -8:

T9 (2, -8) =  t0 + t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8

  =  image78.pngimage79.png

  =  8.679707

When a = 2 and x = -7:

T9 (2, -7)  = image80.pngimage81.png

  = 2.743859

When a = 2 and x = -3:

T9 (2, -3) = image82.pngimage83.png

 = 0.016654

When a = 2 and x = -2:

T9 (2, -2) =  image84.pngimage86.png

= 0.250046

Using Microsoft Excel, I will plot the relation between T9 (2,x) and x (negative) :

x

y

-8

8.679707

-7

2.743859

-3

0.016654

-2

0.250046

image87.png

So, here I haven’t noticed any sign of similarity to the cases before with x-positive. As x increases, Tn decreases then increases. So for the general statement found above, we have to note that it’s untrue for x-negative.

Now, let’s point out a little bit about the scope of the general statement:

image88.png

  • We know the range of values Tn (a,x).
  • We know how Tn (a,x) changes when x changes.
  • We know the domain for a and x: positive numbers.
  • It’s easy to find the infinite sum, just by setting values for a and x.
  • We know this is power series expansion.

What I did to find out this statement is just calculating, and while doing this, I have been realising, step by step, some signs that suggest about the Tn (a,x), like range, sign and simplest formula.

After doing this portfolio, I learned several things, such as

  • using mathematical technology on computer, which I did not know before;
  • constructing the parts of the work so that it looks logically;
  • using appropriate language when doing mathematical big work;
  • realising subtleties from graph/plot rather than from statistics as before I was used to;
  • how to find the sum of infinite general sequence.

I DECLARE THAT THE WHOLE WORK IS ENTIRELY DONE ON MY OWN.

...read more.

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