Divergence of harmonic series

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  1. Why does harmonic series diverge?
  1. Divergence of Harmonic Series
    Show that the following series are divergent:
    1. n=21n \sum_{n=2}^{\infty}\frac{1}{n}
    2. n=15n \sum_{n=1}^{\infty}\frac{5}{n}
    3. n=1[n+1n2] \sum_{n=1}^{\infty}[\frac{n+1}{n^2}]
Topic Notes
In this section, we will talk about the divergence of Harmonic Series. A lot of people think that Harmonic Series are convergent, but it is actually divergent. We will first show a simple proof that Harmonic series are divergent. Then we will tackle some questions which involves algebraically manipulating the series to a Harmonic Series. Note that you can have several cases where some algebraic manipulation can lead to having more series. As long as you show that one of the series is Harmonic, then you can state that the entire thing is divergent.
Note *Harmonic Series are in the form:
n=11n \sum_{n=1}^{\infty}\frac{1}{n}
It is always divergent.