Divergence of harmonic series

Divergence of harmonic series

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.
Basic concepts: Introduction to infinite series,

Lessons

Note *Harmonic Series are in the form:
∑n=1∞1n \sum_{n=1}^{\infty}\frac{1}{n}
It is always divergent.
  • Introduction
    Why does harmonic series diverge?
  • 1.
    Divergence of Harmonic Series
    Show that the following series are divergent:
    a)
    ∑n=2∞1n \sum_{n=2}^{\infty}\frac{1}{n}

    b)
    ∑n=1∞5n \sum_{n=1}^{\infty}\frac{5}{n}

    c)
    ∑n=1∞[n+1n2] \sum_{n=1}^{\infty}[\frac{n+1}{n^2}]

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5.
Sequence and Series
5.1
Introduction to sequences
5.2
Monotonic and bounded sequences
5.3
Introduction to infinite series
5.4
Convergence and divergence of normal infinite series
5.5
Convergence & divergence of geometric series
5.6
Convergence & divergence of telescoping series
5.7
Divergence of harmonic series
5.8
P Series
5.9
Alternating series test
5.10
Divergence test
5.11
Comparison & limit comparison test
5.12
Integral test
5.13
Ratio test
5.14
Root test
5.15
Absolute & conditional convergence
5.16
Radius and interval of convergence with power series
5.17
Functions expressed as power series
5.18
Taylor series and Maclaurin series
5.19
Approximating functions with Taylor polynomials and error bounds
Calculus 2
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