When testing for convergence, how do you determine which test to use?
1 Answer
There is no general method of determining the test you should use to check the convergence of a series.
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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.