Calculus II:

Definite integral
Sequences ans series
Convergence and divergence tests in Series

We denote by {an} the sequence of general term an and by {Sn} the nth partial sum of the series Σ ₁ ^∞

Strategy for Testing Series

We have several ways of testing a series for convergence or divergence, but the problem is to decide which test to use on which series.

The testing series is similar to integrating functions. We have no fixed rules to apply to series to test their convergence. But there are some convenient ways about which test to apply to a given series.

The bad strategy is to apply a list of the tests in a specific order until one finally works. That would be a waste of time and effort. Instead, as with integration, the main strategy is to classify the series according to its form.

1. If lim_{n → ∞} S_n ≠ 0, then the Test for Divergence should be used.

2. If the series is of the form Σ{1/n^p}, it is a p-series, which we know to be convergent if p > 1 and divergent if p ≤ 1 .

3. If the series has the form Σ{arⁿ} or , Σ{ar^{n - 1}, it is a geometric series, which converges if |r| < 1 and diverges if |r| ≥ 1.

Some preliminary algebraic manipulation may be required to bring the series into this form.

4. If the series has a form that is similar to a p-series or a geometric series, then one of the comparison tests should be considered.

In particular, if S_n is a rational function or algebraic function of n (involving roots of polynomials), then the series should be compared with a p-series.

The value of p should be chosen by keeping only the highest powers of n in the numerator and denominator.

The comparison tests apply only to series with positive terms, but if Σ{S_n} has some negative terms, then we can apply the Comparison Test to Σ{|S_n|} and test for absolute convergence.

5. If the series is of the form Σ{(-1)^n-1 b_n} or Σ{(-1)ⁿ b_n, then the Alternating Series Test is an obvious possibility.

6. Series that involve factorials or other products (including a constant raised to the nth power) are often conveniently tested using the Ratio Test.

Bear in mind that |S_n+1/S_n| →1 as n → ∞ for all p-series and therefore all rational or algebraic functions of n. Thus, the Ratio Test should not be used for such series.

7. If S_n = f(n), where ∫₁^∞ f(x) dx is easily evaluated, then the Integral Test is effective.