Question

When testing for convergence, how do you determine which test to use?

Answer 1

There is no general method of determining the test you should use to check the convergence of a series.

• For series where the general term has exponents of `n`, it's useful to use the root test (also known as Cauchy's test).
  Example 1: Power Series
  The definition of the convergence radius of the of a power series comes from the Cauchy test (however, the actual computation is usually done with the following test).

• Generally, the computation of the ratio test (also known as d'Alebert's test) is easier than the computation of the root test.
  Example 2: Inverse Factorial
  For the series `sum_(n=1)^(oo) 1/(n!)` the d'Alembert's test gives us:
  `lim_(n to oo) |1/((n+1)!)|/|1/(n!)| = lim_(n to oo) |n!|/(|(n+1)!|) = lim_(n to oo) |(n!)/((n+1)n!)| = lim_(n to oo) |1/(n+1)| = 0`
  So the series is convergent.

• If you know the result of the improper integral of the function `f(x)` such that `f(n)=a_n`, where `a_n` is the general term of the series being analyzed, then it might be a good idea to use the integral test.
  Example 3: A proof for the Harmonic Series.
  Knowing that the improper integral `int_1^(oo) 1/x dx` is divergent (it's easy to check) implies that the harmonic series `sum_(n=1)^(oo) 1/(n)` diverges.

• Comparision tests are only useful if you know an appropriate series to compare the one you're analyzing to. However, they can be very powerful.
  Example 4: Hyperharmonic Series
  The series of the form `sum_(n=1)^(oo) 1/(n^p)` are called hyperharmonic series or `p`-series. If you can show that the series `sum_(n=1)^(oo) 1/(n^(1+epsilon))` converges, for some small, positive value of `epsilon`, than any `p`-series such that `p>1 + epsilon` converges.