Math Help

Useful Series for Testing of Positive Term Series ∑ a_n

In a Nutshell: The harmonic series, the p-series, and the geometric series are three

Positive term series used to determine convergence or divergence of positive term series.

The tables below detail each of these series.

∞

The harmonic series diverges ∑ b_n = Σ (1/n)

n = 1

The p-series - Converges if p > 1, Diverges if p ≤ 1

∞

Σ b_n = ∑ (1/n^p)

n = 1

The geometric series - Converges to a/(1 – r) if |r| < 1 and diverges if |r| ≥ 1

The geometric series, Σb_n , can be expressed in two forms as follows:,

∞ ∞

∑ (arⁿ)also one can pick n = m – 1, so Σ ar ^m-1 is also the geometric series

_{n = 0
m
= 1}

Strategy: If you can show that the series you are working with is one of these series,

(by comparison) then use it to determine convergence or divergence of your series.

Hint: Select these or suitable variations of these series. i.e. ∑ 1/(n + 1) is a harmonic

series it also diverges

Return to Calculus 2 Notes Involving Sequences, Infinite Series, etc