How do you determine whether the sequence #a_n=ln(ln(n))# converges, if so how do you find the limit?

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Answer 1 · 2017-03-24T12:58:45

The series

#sum_(n=0)^oo ln(ln (n))#

is not convergent.

A necessary condition for any series:

#sum_(n=0)^oo a_n #

to converge is that:

#lim_(n->oo) a_n = 0#

In fact if we consider the #n#-th partial sum:

#s_(n-1) = sum_(k=0)^(n-1) a_k#

we have:

#(1) s_n = s_(n-1) +a_n#

Now if the series is convergent this means that:

#lim_(n->oo) s_n = L# with #L in RR#

and clearly this implies that also:

#lim_(n->oo) s_(n-1) = L#

but as from #(1)#:

#lim_(n->oo) s_n = lim_(n->oo) s_(n-1) + lim_(n->oo) a_n#

we have:

#L = L + lim_(n->oo) a_n#

which implies:

#lim_(n->oo) a_n = 0#

Now as:

#lim_(n->oo) ln(ln (n)) = oo#

the series is not convergent.