Assignment 2

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Exercise 1

The harmonic numbers are defined for n>=1 as

H_n = (1/1) + (1/2) + (1/3) + ... + (1/n)

It is known that as n tends to infinity, the sequence H_n converges to ln(n)+k, where ln is the natural logarithm and k is a constant known as Euler's constant. In this exercise you are asked to compute an approximate value for Euler's constant.

Generate the values of H_n and ln(n) successively for n=1,2,3,...,1000000, and compute the difference H_n - ln(n). Looking at the sequence of these differences, convince yourself that the mathematical theory just mentioned seems to be valid.

You should also convince us about the correctness of your program by showing the results for a small number of representative values of n, for example, for the 100 values n = 10000, 20000, 30000, ..., 1000000. Note that we will be VERY FURIOUS, if we see all the million iterations in your output (which require some twenty thousand pages for printing).

Sample output

Your output should be presented in the following fashion:

n =   10000, H_n =  9.787606036, ln n =  9.210340372, H_n - ln n =  0.577265664
n =   20000, H_n = 10.480728217, ln n =  9.903487553, H_n - ln n =  0.577240665
n =   30000, H_n = 10.886184992, ln n = 10.308952661, H_n - ln n =  0.577232331
n =   40000, H_n = 11.173862898, ln n = 10.596634733, H_n - ln n =  0.577228165
n =   50000, H_n = 11.397003949, ln n = 10.819778284, H_n - ln n =  0.577225665
n =   60000, H_n = 11.579323839, ln n = 11.002099841, H_n - ln n =  0.577223998
n =   70000, H_n = 11.733473329, ln n = 11.156250521, H_n - ln n =  0.577222808
n =   80000, H_n = 11.867003829, ln n = 11.289781914, H_n - ln n =  0.577221915
n =   90000, H_n = 11.984786170, ln n = 11.407564949, H_n - ln n =  0.577221220
n =  100000, H_n = 12.090146130, ln n = 11.512925465, H_n - ln n =  0.577220665

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Exercise 2

Randomly generate a sequence of integers between -5 and +99 and output the maximum and minimum values generated so far. Exit, if a negative integer is generated. You must not store the sequence generated (say using an array), but update the maximum and minimum values on the fly, as soon as a new entry is generated.

Report the output of your program for about five runs with different seeds. At least one of these sequences should be as big as 20+ iterations.

Click here to know how to generate random numbers.

Sample output

Iteration   1: new entry =  84, max =  84, min =  84
Iteration   2: new entry =  87, max =  87, min =  84
Iteration   3: new entry =  72, max =  87, min =  72
Iteration   4: new entry =  53, max =  87, min =  53
Iteration   5: new entry =  93, max =  93, min =  53
Iteration   6: new entry =  23, max =  93, min =  23
Iteration   7: new entry =  88, max =  93, min =  23
Iteration   8: new entry =  52, max =  93, min =  23
Iteration   9: new entry =  17, max =  93, min =  17
Iteration  10: new entry =  46, max =  93, min =  17
Iteration  11: new entry =  61, max =  93, min =  17
Iteration  12: new entry =  16, max =  93, min =  16
Iteration  13: new entry =  70, max =  93, min =  16
Iteration  14: new entry =  37, max =  93, min =  16
Iteration  15: new entry =  19, max =  93, min =  16
Iteration  16: new entry =  33, max =  93, min =  16
Iteration  17: new entry =  60, max =  93, min =  16
Iteration  18: new entry =  36, max =  93, min =  16
Iteration  19: new entry =  84, max =  93, min =  16
Iteration  20: new entry =  70, max =  93, min =  16
Iteration  21: new entry =  40, max =  93, min =  16
Iteration  22: new entry =  28, max =  93, min =  16
Iteration  23: new entry =  36, max =  93, min =  16
Iteration  24: new entry =  96, max =  96, min =  16
Iteration  25: new entry =  54, max =  96, min =  16
Iteration  26: new entry =  76, max =  96, min =  16
Iteration  27: new entry =  17, max =  96, min =  16
Iteration  28: new entry =  -4, ...quitting...

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