HW027
Math 2984 - Fall 2017
Intro to Mathematical Problem Solving

TASK: Given an alternating, decreasing infinite series, add up a finite number of terms that satisfy an accuracy requirement, using a WHILE statement.

COMMENT: An infinite series represents the sum of an infinite sequence of terms, and has the form: 

        result = a1 + a2 + a3 + ... + an + ...
In an alternating infinite series, the odd terms are positive and the even terms are negative, or vice versa, as in: 
        log(2) = 1 - 1/2 + 1/3 - 1/4 + 1/5 - ...
In an alternating infinite series with decreasing terms, each successive term in the series is smaller (in absolute value) than the preceding one. The series for log(2) is an example, since 1/2 is less than 1, 1/3 is less than 1/2, and so on for each term.

There are two important things about such a series:

  1. the series has an exact numeric value, which sometimes can be determined using mathematics;
  2. we can approximate the exact numeric value by adding up the first N terms, and we know that we know that the error in our estimate is no bigger than the size of the next term.

For the log(2) infinite series, this means that the value of log(2) can be estimated by: 

        1,                  plus or minus 1/2
        1-1/2=1/2,          plus or minus 1/3
        1-1/2+1/3=5/6,      plus or minus 1/4
        1-1/2+1/3-1/4=7/22, plus or minus 1/5, and so on

So if we wanted to estimate log(2), with an error no greater than 1/10, we simply add up terms from the series until the next term is smaller than 1/10.

The same idea works for ANY alternating infinite series with terms of decreasing size. We will use this idea to estimate the following infinite series: 

        S = 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - ...
with an accuracy of ACC. The exact value of this infinite series is 1/3.

INSTRUCTIONS: 

        Get a value for the desired accuracy ACC using the input() command.

        Set S_EXACT to 1/3.

        Initialize S, your estimate for the sum of the series, to 0.

        Set A, the value of the "next" term, to 1/2.

        Set N, the number of steps, to 0.

        AS LONG AS the MAGNITUDE of A is bigger than ACC

          increase N

          add A to S

          Compute the next value of A.  You can do this by dividing A by -2,
          or multiplying it by -1/2.

        END

        PRINT S_EXACT
        PRINT S, your estimate for S
        PRINT the actual error, | S - S_EXACT |
        PRINT your estimated error A
        PRINT the requested accuracy ACC
        PRINT the number of terms you added up.

CHECK: Here's an example of results: 

      >> hw027
      Enter desired accuracy: 0.01

      Exact     S =   0.3333333333333333
      Estimated S =   0.3281250000000000
      Error       =   0.0052083333333333
      Error est   =   0.0078125000000000
      Accuracy    =   0.0100000000000000
      Number of terms = 6

SUBMIT: Your work should be stored in a script file called "hw027.m". Your script file should begin with at least three comment lines: 

        % hw027.m
        % YOUR NAME
        % This script (describe what it does)
        % Add any comments here that you care to make.
If this problem is part of an assignment, then submit it to Canvas.