infinite series (idea)

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 Σu_n 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 Σu_n 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 (u_n)
   
   
   • if the sequence (u_n) does not converge toward 0 (no limit or limit different from 0) then Σu_n diverges
     
     
2. Possibly identify Σu_n as a Special Series
   
   
   1. Geometric Series: Σarⁿ
      • if | r | < 1 (r is the ratio of the geometric sequence (arⁿ) ) then Σarⁿ converges (see write-up above to find the limit)
      • if | r | ≥ 1 then Σarⁿ diverges
        
        
   2. p-series: Σ1/n^p (apply p-series theorem)
      • if p > 1 then Σ1/n^p converges
      • if 0 ≤ p ≤ 1 then Σ1/n^p diverges (note: includes harmonic series with p = 1)
        
        
   3. Alternate Series: Σ(-1)ⁿu_n (where (u_n) 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: Σa_nxⁿ
      
      
3. Apply Convergence Tests
   
   
   1. Absolute Convergence: if Σ | u_n | converges, then Σu_n converges (converse is not true).
      
      
   2. Comparison Test: using a sequence (v_n) such as either (u_n ≤ v_n) or (u_n ≥ v_n) 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 Σv_n such as ( | u_n / v_n | ) converges toward a strictly positive value (0 < lim ( | u_n / v_n | ) < +∞).
      • Σu_n is convergent if and only if Σv_n is convergent (see details and example in write-up above).
        
        
   4. Cauchy Condensation Test:
      • if (u_n) is a decreasing sequence of positive terms.
      • then Σu_n is convergent if and only if Σ2^ku_2^k is convergent.
        
        
   5. Cauchy Root Test:
      • (u_n) is a sequence of positive terms and l = lim ( (u_k)^1/k ) (l = ∞, if ( (u_k)^1/k ) diverges)
      • if l < 1 then Σu_n converges
      • if l > 1 (including ∞) then Σu_n 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):
      • (u_n) is a sequence of positive terms and l = lim (u_k+1/u_k)
      • if l < 1 then Σu_n converges
      • if l > 1 (including ∞) then Σu_n 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) = u_n for all n ≥ 1 then Σu_n 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: Σu_n = λΣa_n + γΣb_n 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 Σv_n that will be such as either:
   
   
   • Σv_n convergent and u_n < v_n for all n > N
   • Σv_n divergent and u_{n >} v_n 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).