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In the former articles, we looked at infinite series and specific cases of these: geometric and harmonic series. We were told that harmonic series are always divergent, but we did not show why. In this article, we will see how we can prove it through the so-called Integral Test.
Re-visiting the Harmonic Series
Let us first remember what the notation of the harmonic series is. It looks like the following:
For us to understand why it is divergent, let us first look at the function defined as:
Now, what if we want to find the area under the graph for this function on the interval [1; ∞[? This is the same as taking the integral of our function from 1 to infinity. It would look like so:
We know that the integral diverges since it gives an area of infinity. The question is, of course, how that helps us with our harmonic series. We then have to recall a knowledge we have about integration — we estimate an area under a graph by breaking up the interval into segments and then sketching in rectangles. We then find the sum of the area of all these rectangles as an estimate of the actual area under the graph. Let us see it visually for our function above:
We can then say that the area under the graph is approximative:
