P Series

P Series

In this lesson, we will learn about p-series. They take on a special form, and look very similar to Harmonic series. However their convergence or divergence depends on the denominator's exponent, p. If p is greater than 1, then the series converge. If p is less than 1, then the series diverge. In this lesson, we will start off with looking at some simple p-series questions. Then we will look at a complicated p-series which convergences and divergences depending on a certain value.

Lessons

Note *P Series are in the form:
n=11np \sum_{n=1}^{\infty}\frac{1}{n^p}
where if pp > 1 then the series converge. Otherwise, the series diverges.
  • Introduction
    P series Overview

  • 1.
    Convergence and Divergence of P Series
    Determine whether the series is convergent or divergent
    a)
    n=31n2 \sum_{n=3}^{\infty}\frac{1}{n^2}

    b)
    n=1n3+1n2 \sum_{n=1}^{\infty}\frac{n^3+1}{n^2}


  • 2.
    For what values of kk does the series n=1n3+1n(2k+1)\sum_{n=1}^{\infty}\frac{n^3+1}{n^{(2k+1)}} converge and diverge?