Absolute and Conditional Convergence

Varying Signed Series :

When we have a series with terms that are not strictly positive but are not alternating either, we test it for absolute convergence. The most common test is the ratio test applied to the absolute value of the sum of terms.

Absolute Convergence:

An infinite series is absolutely convergent if the series of absolute values is convergent. Generally, we use the Ratio Test on the absolute value of the series terms, to show that a series is absolutely convergent, however, we can use any of the convergence tests on it.

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Example: the series includes both positive and negative terms but it is not alternating, so we test. Since we know that , and we know that n² is always > 0, we can compare it to , a convergent p-series with p = 2. By the basic comparison test, this series is absolutely convergent.

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Conditional Convergence :

Some series that include both positive and negative signs can be shown to be convergent in one form but divergent in another. The classic example is the harmonic series which is divergent as a positive term series but convergent as an alternating series .

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Example: use the Ratio test to check for absolute convergence.

Solution: We take limit at infinity of the (n + 1) term divided by the n-th term like this.

now divide through by n², to get a limit of ½. Since ½ < 1, the series is absolutely convergent.

Practice:

Test for Absolute or Conditional Convergence. State the test you use.

 1) 2) 3) 4) 5) 6)

Solutions

Notes:

AST = Alternating Series Test, BCT = basic comp. test, LCT = limit comp. test
RT = ratio test, RtT = root test, IT = Integral Test, TD = test for divergence.
AS = alternating Series, GS = Geometric Series, PS = p-series.
AC = absolutely convergent, CC = conditionally convergent, D = divergent.

Before we use the AST or Alternating Series Test, we must show that the terms are decreasing. In all cases in this exercise, we would simply show that the denominator grows huge whereas the numerator either remains constant or remains smaller than the denominator. For this reason, the statement of decreasing terms has been omitted in the solutions when the AST was applied.

 1) AST: lim = 0 so converges as ASBCT: use , D p-seriessince an > bn , series diverges.Series is CC 2) AST: lim = 0 so converges as ASPS: p-series, p = 2/3 < 1 so D.Series is CC. 3) TD makes series Divergent. 4) AST: lim = 0 so converges as ASBCT: use , C p-seriessince an < bn , series is AC. 5) AST: lim = 0 so converges as ASRT: series is AC. 6) RT:series Diverges

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