Absolute & conditional convergence

Absolute & conditional convergence

In this section, we will learn about the concept of absolute and conditional convergence. We say a series is absolutely convergent if BOTH the series and absolute value of the series is convergent. If the series is convergent and the absolute value of the series is divergent, then we call that conditional convergence. First, we will be use these definitions and apply it to some of the series below. Lastly, we will look at a complicated series which requires us to convert it to a simpler form before showing if it's absolutely convergent, conditionally convergent, or divergent.

Lessons

Let an\sum a_n be a convergent series. Then we say that an\sum a_n is absolutely convergent if an\sum |a_n| is convergent.

If an\sum |a_n| is divergent, then we say that an\sum a_n is conditionally convergent.

  • Introduction
    Absolute & Conditional Convergence Overview

  • 1.
    Questions based on Absolute & Conditional Convergence
    Determine if the series is absolutely convergent, conditionally convergent, or divergent
    a)
    n=2(1)nn1 \sum_{n=2}^{\infty}\frac{(-1)^n}{n-1}

    b)
    n=1(1)nn2 \sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}

    c)
    n=22+cosnn \sum_{n=2}^{\infty}\frac{2+cosn}{n}

    d)
    n=4(n2+2)(1)3n+1(n4+1)1n1 \sum_{n=4}^{\infty}\frac{(n^2+2)(-1)^{3n+1}}{(n^4+1)1^{n-1}}


  • 2.
    Advanced Question
    Determine if the series n=1(1)n2sin2((2n+1)π2)n3 \sum_{n=1}^{\infty}\frac{(-1)^{n-2}sin^2(\frac{(2n+1)\pi}{2})}{n^3} is absolutely convergent, conditionally convergent, or divergent.