Absolute & Conditional Convergence (College Board AP® Calculus BC)

Absolute & conditional convergence

What is absolute convergence?

• Let be an infinite series

  • If , converges, then is said to converge absolutely

  • I.e. a series is absolutely convergent if the corresponding series of absolute values converges

• If a series converges absolutely then it converges

  • I.e. if you know that converges, that tells you that converges as well

    • There is no need to test separately for convergence

What is conditional convergence?

• Let be an alternating series (i.e. with alternating positive and negative terms)

  • If converges, but does not converge, then is said to converge conditionally

  • I.e. a series is conditionally convergent if the series itself converges, but the corresponding series of absolute values does not

• Note that if does not converge, that doesn't tell you anything about the convergence of

  • I.e., may converge or diverge, and needs to be tested separately

Why is the difference between absolute and conditional convergence important?

• If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms in the series has the same value

• This is probably what you are used to when adding numbers

  • I.e. rearranging the order or regrouping (by adding or removing brackets) doesn't change the sum

  • But this is only true when adding together finitely many numbers

    • It is not necessarily true for an infinite series!

• For example the series

  • This converges to a sum of , as can be seen by writing it as

    

  • It is conditionally convergent because the sum of absolute values gives

    

    • That is 2 times the harmonic series, which we know diverges to

  • However the original series can be rearranged and regrouped to give a different sum

    

    • With that rearrangement it is equal to the alternating harmonic series, which converges to a sum of

    • This appears to contradict it converging to above

• Conditionally convergent series can be regrouped and/or rearranged to give

  • a series with a different sum

  • or a series which does not converge to any sum