Math Help

Positive Term Infinite Series ∑ a_n Basics and Ones You Need to Know, Tests

In a Nut Shell: Why consider infinite Series? Not all functions of interest to engineers.

are integrable i.e. exp(-x²) Also it may be useful to represent a function by the first

few terms of an infinite series in order to obtain approximate values of the function.

Basics: There are basically four types of infinite series to consider. They are

positive term series, alternating term series, product series, and telescoping series. i.e.

∑ (1/n) ∑ (˗1)ⁿ (1/n) ∑ [ 2 • 4 • 6 • • • (2n) ] / n! ∑ 1 / n(n+1)

The major interests in the study of infinite series are to (1) determine if the series

converges or diverges and (2) to determine the sum of the series. In most cases

determining the sum of the series is beyond the scope of undergraduate calculus.

The primary focus is then to determine whether the infinite series converges or diverges.

Positive term series used for evaluation of convergence or divergence (must know):

The harmonic series diverges

∞

∑ (1/n)

_{n = 1}

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

∞

∑ (1/n^p)

_{n = 1}

The geometric series - Converges if |r| < 1 and diverges if |r| ≥ 1

If the geometric series converges its sum, S, is S = a/(1 – r) .

∞ ∞

∑ (arⁿ)also one can pick n = m – 1, so Σ ar ^m-1 is also the geometric series
n=0 n=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.

Mathematicians have developed a number of theorems to determine if infinite series

converge or diverge. Some theorems only apply to positive terms series, some apply

only to alternating term series, and some apply to both types of series.

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