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