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Positive Term Series - Theorem 1


By a positive term series, we mean a series in which every term is greater than zero. For example, the geometric series

1 + c + c2 + ... + cn + ...

is a positive term series if c > 0 but not if c ≤ 0. We call a sequence S1, S2, ..., Sn, ... increasing if Sm < Sn whenever m < n. It is easy to see that

at + a2 + ... + an + ...

is a positive term series if and only if its partial sum sequence is increasing. We are going to give several tests for the convergence of a positive term series. The starting point is the following theorem.


An increasing sequence <Sn> either converges or diverges to ∞.

Geometrically, this says that, as n gets large, the graph of the sequence either levels out at a limit L or the value of Sn gets large (Figure 9.4.1). We omit the proof. (The proof is given in the Epilogue at the end of the book.)


Figure 9.4.1

Theorem 1 has an equivalent form for positive term series because the partial sum sequence of a positive term series is increasing.

THEOREM 1 (Second Form)

A positive term series either converges or diverges to ∞.

Example 1: Harmonic Series Diverges to ∞
Example 2: Geometric Series

Remark: Theorem 1 shows that to determine whether a positive term series converges, we need only look at one infinite partial sum. If it is finite the series converges and if it is infinite the series diverges to ∞.

Last Update: 2006-11-07