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Determining the convergence of a series

Because there are only a (relatively) small number of theorems applicable toward determining the convergence of infinite series, it is quite possible to define an "algorithmic" solving process to the question: Does Σun converge/diverge?

Note the use of quotes around the word algorithmic above: it is there to stress the fact I won't be using any formal meta-notation and that I am not putting any claim on fulfilling actual algorithm requirements in this method. It is merely an attempt at organizing all the common tests and tools available to deal with series into a coherent and easy to use list of "if, then" statements.

Anyway, let's say you got an infinite series Σun and want/need/hope to determine whether it is convergent or divergent. Here is probably how you should go at it:

  1. Check for Essential Condition on (un)

    • if the sequence (un) does not converge toward 0 (no limit or limit different from 0) then Σun diverges

  2. Possibly identify Σun as a Special Series

    1. Geometric Series: Σarn
    2. p-series: Σ1/np (apply p-series theorem)
      • if p > 1 then Σ1/np converges
      • if 0 p ≤ 1 then Σ1/np diverges (note: includes harmonic series with p = 1)

    3. Alternate Series: Σ(-1)nun (where (un) is positive, decreasing and converges to 0) converges

    4. Euler Series: Σ1 / n! converges to e (convergence can be proven easily with d'Alembert's ratio test, finding the limit is a whole other problem)

    5. Power Series: Σanxn

  3. Apply Convergence Tests

    1. Absolute Convergence: if Σ | un | converges, then Σun converges (converse is not true).

    2. Comparison Test: using a sequence (vn) such as either (un vn) or (un vn) for all n > N (see details in write-up above).

    3. Limit Comparison Test (same principle as the Comparison Test, but easier to apply):
      • judiciously define a series Σvn such as ( | un / vn | ) converges toward a strictly positive value (0 < lim ( | un / vn | ) < +∞).
      • Σun is convergent if and only if Σvn is convergent (see details and example in write-up above).

    4. Cauchy Condensation Test:
      • if (un) is a decreasing sequence of positive terms.
      • then Σun is convergent if and only if Σ2ku2k is convergent.

    5. Cauchy Root Test:
      • (un) is a sequence of positive terms and l = lim ( (uk)1/k ) (l = ∞, if ( (uk)1/k ) diverges)
      • if l < 1 then Σun converges
      • if l > 1 (including) then Σun diverges
      • if l = 1 then Brian only knows (the test is inconclusive, keep looking)

    6. D'Alembert's Ratio Test ("weaker" than Cauchy Root Test but easier to apply):
      • (un) is a sequence of positive terms and l = lim (uk+1/uk)
      • if l < 1 then Σun converges
      • if l > 1 (including) then Σun diverges
      • if l = 1 then the test is inconclusive

    7. Integral Test: if you can find a positive, continuous, decreasing function f such as f(n) = un for all n ≥ 1 then Σun is convergent if and only if 1+∞f(x)dx is convergent (see details in write-up above)

    8. Abel's Test is a sophisticated test mostly used to prove Alternate Series convergence (unlikely to be useful in most cases)

  4. If the tests provided in step 3. do not yield any direct results, try splitting the series in a linear combination of series: Σun = λΣan + γΣbn and try again on each of the series

  5. If you still cannot conclude, your best chances stand with using the Comparison test and move the problem into solving the convergence of a series Σvn that will be such as either:

    • Σvn convergent and un < vn for all n > N
    • Σvn divergent and un > vn for all n > N

  6. If you still cannot conclude, try banging your head repeatedly against the nearest wall and start again from step 1. (it might work sometimes, provided you do not go too hard on the banging)

  7. If you still cannot conclude, consider (in no particular order of preference and depending on possibilities):

    1. suicide
    2. murder
    3. career change
    4. turning the page of your textbook

Although I would love to believe that the above list is somewhat close to exhaustive, it is most likely very not so. Out of ignorance, stupidity, plain laziness or a combination of the three, I might have left out useful tests or ideas. If you know of any beneficial addition to this methodology please let me know (try remaining in the strict field of infinite series, though, as I'm not trying to rewrite Principia Mathematica here).