P Series - Sequence and 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.

Basic concepts:
Introduction to infinite series
Divergence of harmonic series

Lessons

Notes:

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

1.
Convergence and Divergence of P Series
Determine whether the series is convergent or divergent

P Series

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CalculusIntroduction to infinite series

CalculusDivergence of harmonic series

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Notes:

Note *P Series are in the form: ∑n=1∞1np \sum_{n=1}^{\infty}\frac{1}{n^p}∑​n=1​∞​​​n​p​​​​1​​ where if ppp > 1 then the series converge. Otherwise, the series diverges.

Intro Lesson

P series Overview

1.

Convergence and Divergence of P Series Determine whether the series is convergent or divergent

a)

∑n=3∞1n2 \sum_{n=3}^{\infty}\frac{1}{n^2}∑​n=3​∞​​​n​2​​​​1​​

b)

∑n=1∞n3+1n2 \sum_{n=1}^{\infty}\frac{n^3+1}{n^2}∑​n=1​∞​​​n​2​​​​n​3​​+1​​

2.

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

P Series

Don't just watch, practice makes perfect.

We have over 350 practice questions in Calculus for you to master.

Calculus
Introduction to infinite series

Calculus
Divergence of harmonic series

or

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

Convergence and Divergence of P Series
Determine whether the series is convergent or divergent

$\sum_{n=3}^{\infty}\frac{1}{n^2}$

$\sum_{n=1}^{\infty}\frac{n^3+1}{n^2}$

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