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Positive Term Series  Theorem 1
9.4 SERIES WITH POSITIVE TERMS By a positive term series, we mean a series in which every term is greater than zero. For example, the geometric series 1 + c + c^{2} + ... + c^{n} + ... is a positive term series if c > 0 but not if c ≤ 0. We call a sequence S_{1}, S_{2}, ..., S_{n}, ... increasing if S_{m} < S_{n} whenever m < n. It is easy to see that a_{t} + a_{2} + ... + a_{n} + ... 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. THEOREM 1 An increasing sequence <S_{n}> 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 S_{n} 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 ∞.
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 ∞.


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