• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12

# 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

 002329/

Anglo-Chinese School (Independent)

INFINITE SUMMATION

“I undersigned, hereby declare that the following course work is all my own work and that I worked independently on it”

__________________

Name:                                Pranav Sharma _______________

Class:                                5.16 Ephesians_______________

Index Number:                ____________________________

Subject Teacher:                ____________________________

Introduction

• Technology used:
• Microsoft word 2010
• Microsoft Excel 2010

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.

Let

, therefore I will calculate the

of the first 10 terms of the above sequence for

After I calculated the sums for

I am going to represent the relation between

on a graph:

 n 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

From this graph we can see that as

, the value of n increases as well, but it does not exceed 2, thus the greatest value of this relationship will be 2 and therefore the domain of this relationship is

, as n approaches

.

Let

therefore the sequence of

1.  will look in the following way:

After I calculated the sums for

I am going to represent the relation between

on a graph:

 n 1 2,098612 2 2,702086 3 2,923080 4 2,983776 5 2,997112 6 2,999553 7 2,999936 8 2,999988 9 2,999994 10 2,999994

From this graph we can see that as

, the value of n increases as well, but it does not exceed 3, thus the greatest value of this relationship will be 3 and therefore the domain of this relationship is

, as n approaches

.

Middle

4

4,878967

5

4,968956

6

4,993095

7

4,998645

8

4,999761

9

4,999961

10

4,999993

From this graph we can see that as

, the value of n increases as well, but it does not exceed 5, thus the greatest value of this relationship will be 5 and therefore the domain of this relationship is

, as n approaches

.

Let

therefore the sum of

will look the following way:

After I calculated the sums for

I am going to represent the relation between

on a graph:

 n 1 2,945910 2 4,839193 3 6,067246 4 6,664666 5 6,897171 6 6,972577 7 6,993539 8 6,998637 9 6,999740 10 6,999954

From this graph we can see that as

, the value of n increases as well, but it does not exceed 7, thus the greatest value of this relationship will be 7 and therefore the domain of this relationship is

, as n approaches

.

From all of the calculations which I performed now I can find a general statement for finding the sequence of this sum (

). In order to continue with finding the general formula, first I have to simplify the original sum of infinite sequences:

. In order to do this I will replace (xlna) with m, therefore this sequence will look in the following:

Hence the sum of these infinite terms will be:

However on power series expansion’s side my general statement will have the following form:

From here follows that the general statement for the infinite sequences is

I concluded that this is the general statement, on the basis of the following facts: as

increases the value of n will increase also

Now I want to expand my investigation in order to find a general statement which represents the infinite sum of the infinite sequence

, where

In order to find the general statement of these infinite sequences I will define

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

First I will find

, where I have a=2 and I calculate x for 1,2,3,4,5,6,7,8,9,10

Therefore I will calculate the sum for the first nine terms:

In order to comment on the relation between

2x and x I need to plot on a graph:

 x (2,x) 1 1,999999 2 3,999939 3 7,997485 4 15,963511 5 31,702131 6 62,305349 7 120,465724 8 227,963677 9 502,560081 10 755,692613

Looking at the graph I can conclude that as x increases, the value of

increases as well.

Now I am going to perform the same calculations as above, but with values of a=3 and x=1,2,3,4,5,6,7,8,9,10

Therefore I will calculate the sum for the first nine terms:

In order to comment on the relation between

(3,x) and x I need to plot on a graph:

 X (2,x) 1 2,999992 2 6,798587 3 26,814822 4 78,119155 5 217,471546 6 569,033836 7 1969,842574 8 3060,893218 9 6802,981103 10 13771,38794

Conclusion

 x=2 x=3 x=4 x=8 x=6 a=2 4 8 16 64 256 a=3 9 27 81 729 6561 a=4 16 64 256 4096 65536 a=5 25 125 625 15625 390625 a=6 36 216 1296 46656 1679616

From the table which I made I can clearly see that as value of a and x increases the value of

increases also, which support my general statement and proves the statement that as the value of

, the values of a and x increases too. Therefore my general statement can fit any values of a and x.

• 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.

Page  of

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## This portfolio is an investigation into how the median Body Mass Index of a ...

height ratio will have such drastic changes every 18 years (one period of the sinusoidal function graph). Using the logistic regression function, the BMI of a 30-year-old woman in the US in 2000 would be 22.7. Using the same assumption that human growth is related to the measurable value of

2. ## Stellar Numbers. After establishing the general formula for the triangular numbers, stellar (star) shapes ...

Term Number (n) 1 2 3 4 5 6 Stellar Number (Un) 1 7 19 37 61 91 As seen in the diagram above, the second difference is the same between the terms, and the sequence is therefore yet again quadratic.

1. ## This essay will examine theoretical and experimental probability in relation to the Korean card ...

Nan-Cho, June is Mo- Ran, July is Hong-Ssa-Ri, August is Gong-San, September is Kook-Jun, October is Dan-Feng, November is Oh-Dong and December is Bi. This game "Sut-Da" does not use all these 48 cards but only uses 20 cards. This game can hold two to ten people in one game,

2. ## Mathematics Portfolio. In this portfolio project, the task at hand is to investigate the ...

Three significant figures are taken of both constants to produce the equation: +Gx=11.3t-.250. Regression model using graphing package of: +Gx =11.3t-.250 (scale: 35 min x 40 g) Like the function drawn by hand, the function illustrates that as +Gx decreases, the time that humans can tolerate the force increases.

1. ## Infinite Summation Internal Assessment ...

This table shows the large changes in the value of throughout, while quickly reaching the horizontal asymptote of 128, but this horizontal asymptote is not visible in the small values of . This graph displays = 2 and = 7 and, which creates the horizontal asymptote at 128 as is 128.

2. ## Population trends. The aim of this investigation is to find out more about different ...

This equation will be a way to find out what the value of is, this way it will be possible to graph the equation. The time can be considered as so it can be graphed, as stated at the start of this investigation, the represent the year since 1950.

1. ## SL Type 1 PF - Infinite Summation - A general statement has been reached, ...

For , the value of approaches 2, approaches 4, 8, and so on. This draws a hint about the general pattern of the sum of an infinite sequence when the value is fixed. In order to visualize the values for an analysis of a pattern, Figure 4 plots the values on a graph.

2. ## While the general population may be 15% left handed, MENSA membership is populated to ...

Evidence of Research I conducted my research in the form of a poll on the social networking website Facebook.com. After creating the poll on the website, it was posted where all of my Facebook friends were able to see it and answer in a quick and convenient manner.

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to