EASY PROOF the HARMONIC SERIES diverges! | FUN Math for ALL
We prove in a simple way (you only need to know how to add fractions and what are inequalities) that the harmonic series (in the thumbnail) diverges - which means that its sum is ∞. We make precise what this means in the language of rigorous math - it simply means that no matter how large a number we have, we can add up enough terms in this series to exceed that number. We then make observations step by step and exhibit how adding up more and more numbers in the harmonic series really can get very large. Finally, we summarize our observations to establish a rigorous mathematical proof that the harmonic series diverges. We prove that the sum of the first 2^n terms in the harmonic series is at least 1 + n/2 (a lower bound that gets arbitrarily large as n gets arbitrarily large). We also highlight the important concept of logarithmic growth in this example.
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