Convergence and divergence of normal infinite series

Convergence and divergence of normal infinite series

In this section, we will take a look at normal infinite series that can be converted into partial sums. We will start by learning how to convert the series into a partial sum, and then take the limit. If we take the limit as n goes to infinity, then we can determine if the series is converging or diverging. Note that not all series can be turned into a partial sum. In that case, you would have to use other methods to see if the infinite series is convergent or divergent.
Basic concepts: Introduction to infinite series,
Related concepts: Arithmetic series, Infinite geometric series,

Lessons

  • Introduction
    Overview of Converging and Diverging Series
  • 1.
    Converging and Diverging Series with the formula of partial sums

    You are given the general formula of partial sums for the following series. Determine whether the series converges or diverges.
    a)
    ∑n=1Nan=N2+2N+3N+6 \sum_{n=1}^{N}a_n=\frac{N^2+2N+3}{N+6}

    b)
    ∑n=1Nan=N2+6N+2N2+4 \sum_{n=1}^{N}a_n=\frac{N^2+6N+2}{N^2+4}

    c)
    ∑n=1Nan=N+5N2+1 \sum_{n=1}^{N}a_n=\frac{N+5}{N^2+1}

  • 2.
    Converging and Diverging Series without the formula of partial sums

    Determine whether the following series converges or diverges.
    a)
    ∑n=1∞n \sum_{n=1}^{\infty}n

    b)
    ∑n=1∞n3 \sum_{n=1}^{\infty}n^3

    c)
    2+4+6+8+ ...

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28.
Series
28.1
Arithmetic series
28.2
Geometric series
28.3
Convergence and divergence of normal infinite series
28.4
Convergence & divergence of geometric series
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