Preview

Infinite Summation

Satisfactory Essays
Open Document
Open Document
1462 Words
Grammar
Grammar
Plagiarism
Plagiarism
Writing
Writing
Score
Score
Infinite Summation
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

You May Also Find These Documents Helpful

  • Good Essays

    Math 533

    • 16737 Words
    • 67 Pages

    0.02 0.0080 0.0478 0.0871 0.1255 0.1628 0.1985 0.2324 0.2642 0.2939 0.3212 0.3461 0.3686 0.3888 0.4066 0.4222 0.4357 0.4474 0.4573 0.4656 0.4726 0.4783 0.4830 0.4868 0.4898 0.4922 0.4941 0.4956 0.4967 0.4976 0.4982 0.4987…

    • 16737 Words
    • 67 Pages
    Good Essays
  • Satisfactory Essays

    Pre Calc Checkup 3

    • 240 Words
    • 4 Pages

    For problems 6 – 10, describe the end behavior for each of the functions by writing a limit expression.…

    • 240 Words
    • 4 Pages
    Satisfactory Essays
  • Good Essays

    Problems1

    • 605 Words
    • 2 Pages

    guess, use MATLAB/Octave to make a plot of log(cond(Vn )) against n. You will need to…

    • 605 Words
    • 2 Pages
    Good Essays
  • Better Essays

    Mat 126 Week 1 Assignment

    • 1084 Words
    • 5 Pages

    “An arithmetic sequence is a sequence of numbers in which each succeeding term differs from the preceding term by the same amount. This amount is known as the common difference.” (Bluman, A. G. 2500, page 221)…

    • 1084 Words
    • 5 Pages
    Better Essays
  • Satisfactory Essays

    Kilogram and Question

    • 593 Words
    • 3 Pages

    Quiz 2: Questions and answers from week 2 covering Chapters 3 and 5 in Cracolice and Peters. Question 1: Which of the following numbers are correctly expressed in exponential notation? i. 43,200,000 = 4.32 x 10 7 ii. 0.000977 = 9.77 x 10 4 iii. 606,000 = 6.06 x 10 -5 iv. 0.00000175 = 1.75 x 10 -6 Answer: i and iv Question 2: Which of the following numbers are correctly expressed in ordinary decimal form? i. 6.21 x 10 5 = 621,000 ii. 7.28 x 10 6 = 0.00000728 iii. 9.03 x 10 -4 = 0.000903 iv. 1.12 x 10 -3 = 1120 Answer: i and iii Question 3: Complete the following operation: 8.36 x 106 + 1.320 x 107 Answer: 2.156 x 107 Question 4: Complete the following operation:…

    • 593 Words
    • 3 Pages
    Satisfactory Essays
  • Good Essays

    We divide the numbers and we get .2472 * 10-2. Now we simplify the scientific notation so we end with:…

    • 1300 Words
    • 6 Pages
    Good Essays
  • Satisfactory Essays

    Name________________________________ UDC – Quantitative Reasoning I EXAMINATION 3 – Personal Finance – Exponential Functions Fall - 2012 Instructions: This exam is worth 100 points. Read each question carefully. Answer each question clearly and concisely. Please, show All of Your Work. Remember, I do not believe in magic!!!…

    • 588 Words
    • 3 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Ap Calculus Essay

    • 3066 Words
    • 13 Pages

    = . As x → ∞ , those x 5 terms are like gymnasiums full of sand.…

    • 3066 Words
    • 13 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Managerial MIdterm

    • 638 Words
    • 3 Pages

    Note, this is a closed book examination. Use only simple calculators; smart calculators, smart phones and cell phones cannot be used. There are three questions. Good luck.…

    • 638 Words
    • 3 Pages
    Satisfactory Essays
  • Better Essays

    Beetroot Lab Report

    • 1395 Words
    • 6 Pages

    Table one; shows the effect of water temperature has on the rate at which the anthocyanin pigment leaves of a beetroot tap root cells…

    • 1395 Words
    • 6 Pages
    Better Essays
  • Satisfactory Essays

    Square Roots And Cube Roots

    • 115 Words
    • 146 Pages

    No. Square. Cube. Sq.…

    • 115 Words
    • 146 Pages
    Satisfactory Essays
  • Satisfactory Essays

    1 6. 1 7. 1 8. 1 9. 1 10. 2 11. 2 12. 2 13.…

    • 2040 Words
    • 9 Pages
    Satisfactory Essays
  • Satisfactory Essays

    Chapter 2, Module 2 Notes

    • 660 Words
    • 3 Pages

    * nth root of 1.17 x 1.08 x 1.02 x 1.15 – 1 = 10.34…

    • 660 Words
    • 3 Pages
    Satisfactory Essays
  • Good Essays

    Correlation Analysis

    • 3446 Words
    • 14 Pages

    The problem of analyzing the relation between different series should be broken down nto 3 steps:…

    • 3446 Words
    • 14 Pages
    Good Essays
  • Satisfactory Essays

    Physics Lab Report

    • 563 Words
    • 3 Pages

    In this project we attempted to confirm that the acceleration due to gravity of a freely falling body is…

    • 563 Words
    • 3 Pages
    Satisfactory Essays