The root test states that given a series with positive terms:
#sum_(n=0)^oo a_n# with #a_n >=0#
if the succession #{root(n)(a_n)}# is convergent:
#lim_(n->oo) root(n)(a_n) = L#
then we have:
#0 <=L < 1 => sum_(n=0)^oo a_n# is convergent
# L > 1 => sum_(n=0)^oo a_n =oo#
If #L = 1# then the test does not give us any information.
In fact, suppose that:
#lim_(n->oo) root(n)(a_n) = L < 1#
this means that for any #epsilon > 0# we can find #N# such that:
#root(n)(a_n) < L+epsilon# for #n > N#
As # L < 1# we can choose #epsilon# such that:
#L+epsilon < 1#
Then we have, for #n > N#:
#root(n)(a_n) < L + epsilon < 1#
and elevating both sides to the #n#-th power, which preserves the direction of the inequality:
#a_n < (L+epsilon)^n#
Now:
#sum_(n=0)^oo (L+epsilon)^n#
is a geometric series of ratio #L+epsilon < 1# and is absolutely convergent, so also:
#sum_(n=0)^oo a_n#
is convergent by direct comparison.
In the same way if #L > 1# we can establish the inequality:
#a_n > (L - epsilon)^n# with #L-epsilon > 1#
and determine that #sum_(n=0)^oo a_n# is divergent by direct comparison with a divergent geometric series.
