{"id":4499,"date":"2020-12-19T00:01:05","date_gmt":"2020-12-19T08:01:05","guid":{"rendered":"https:\/\/c-for-dummies.com\/blog\/?p=4499"},"modified":"2020-12-19T08:53:24","modified_gmt":"2020-12-19T16:53:24","slug":"harmonic-series-divergence-and-c-code","status":"publish","type":"post","link":"https:\/\/c-for-dummies.com\/blog\/?p=4499","title":{"rendered":"Harmonic Series, Divergence, and C Code"},"content":{"rendered":"

A harmonic series is a mathematical contraption that deals with cascading fractions. Like the Fibonacci series, I thought I could easily code a harmonic series in C — which I did, but not before reading up on the topic of divergence.
\n
\nPicking up from last week’s Lesson<\/a>, a harmonic series is shown in Figure 1, where it represents the total of a series of smaller and smaller fractions. What is the total?<\/p>\n

\"A

Figure 1. A harmonic series illustrated in a most frightening manner.<\/p><\/div>\n

Never mind doing the math! What I did was to write computer code that made the calculations for me, adding the values 1\/1, 1\/2, on up through 1\/999. Here is the code:<\/p>\n

2020_12_19-Lesson-a.c<\/a><\/h3>\n
\r\n#include <stdio.h>\r\n\r\nint main()\r\n{\r\n    int x;\r\n    float t;\r\n\r\n    t = 1.0;\r\n    for( x=2; x<=1000; x++ )\r\n    {\r\n        t += 1.0 \/ (float)x;\r\n    }\r\n    printf(\"The sum of the harmonic series is %.4f\\n\",t);\r\n\r\n    return(0);\r\n}<\/pre>\n
The for<\/em> loop at Line 9 does a sum of the first 1,000 items in the harmonic series. Each spin of the loop adds another fraction to the total, from 1\/1 to 1\/999. After all these spins, here is the output:<\/p>\n
The sum of the harmonic series is 7.4855<\/code><\/p>\n
I felt rather good about myself for performing this calculation, though I then proceeded to increase the looping value from 1,000 to 1,000,000. Here is the new output:<\/p>\n
The sum of the harmonic series is 14.3574<\/code><\/p>\n
Whoa! It’s getting bigger.<\/p>\n
Before I increased the looping value further, I wondered what the sum actually was. So I watched the rest of the video on harmonic series<\/a> only to discover that the series is divergent. Which means that its sum is infinity.<\/p>\n
The divergence concept was surprising to me. After studying Zeno’s paradoxes, specifically Achilles and the tortoise, I would think that a harmonic series would flatten out at a certain point. But no! Many proofs abound to show that the series is divergent and continues to grow to infinity.<\/p>\n
In fact, in the video, it states that to reach the value of 100.0, my code must loop 15.092×1042<\/sup> times. It would take a computer days to perform this calculation. And somehow it eventually gets to infinity.<\/p>\n
Not every series climbs to infinity. In fact, the sum of a series of powers-of-two fractions flattens out to the value 2.0. This series is illustrated in Figure 2.<\/p>\n
\"An
Figure 1. An power-of-two series thing.<\/p><\/div>\n
My code to generate a power-of-two series hinges upon a for<\/em> loop:<\/p>\n
for( x=2; x<=1000; x*=2 )<\/code><\/p>\n
This loop generates x<\/code> values as multiples of 2: 2, 4, 8, 16, and so on. An expression within the loop calculates the sum of fractions 1+1\/2+1\/4… as illustrated in the Figure.<\/p>\n
2020_12_19-Lesson-b.c<\/a><\/h3>\n
\r\n#include <stdio.h>\r\n\r\nint main()\r\n{\r\n    int x;\r\n    float t;\r\n\r\n    t = 1.0;\r\n    for( x=2; x<=1000; x*=2 )\r\n    {\r\n        t += 1.0 \/ (float)x;\r\n        printf(\"+1\/%d = %.4f\\n\",x,t);\r\n    }\r\n    printf(\"The sum of the even series is %.4f\\n\",t);\r\n\r\n    return(0);\r\n}<\/pre>\n
Line 11 calculates the sum of the powers-of-two series, updating the value in variable t<\/code> as it goes. Because the values increase rapidly, the code outputs the equations and running total at Line 12. The final total is output at Line 14:<\/p>\n
+1\/2 = 1.5000
\n+1\/4 = 1.7500
\n+1\/8 = 1.8750
\n+1\/16 = 1.9375
\n+1\/32 = 1.9688
\n+1\/64 = 1.9844
\n+1\/128 = 1.9922
\n+1\/256 = 1.9961
\n+1\/512 = 1.9980
\nThe sum of the even series is 1.9980<\/code><\/p>\n
Unlike the harmonic series, which grows to infinity, this series approaches 2.0. It is non-divergent, and code easily proves so. None of this would otherwise be interesting to me, but because I can code it in C, it becomes a fun little challenge.<\/p>\n","protected":false},"excerpt":{"rendered":"
My inner math nerd comes out again as I explore the concepts of harmonic series, which work better for me in C code than in maths class. Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-4499","post","type-post","status-publish","format-standard","hentry","category-main"],"_links":{"self":[{"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/4499","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4499"}],"version-history":[{"count":9,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/4499\/revisions"}],"predecessor-version":[{"id":4533,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/4499\/revisions\/4533"}],"wp:attachment":[{"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4499"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4499"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/c-for-dummies.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4499"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}