- [Phillip] So we want to determine
if the following series
converges or diverges,
first thing we notice is
this piece right here,
this is in fact an alternating series.
So we're gonna run our
alternating series test,
so number one, let's check that the limit
as n goes to infinity of one
over square root of n plus 2,
well this is gonna grow,
this is gonna go to infinity,
so one over that infinite
growth is gonna be zero.
So that's check.
Two, the other thing, a sub n plus one,
is equal to one over square
root of n plus one, plus two.
Now we know that square root of n plus one
is greater than square root of n.
Therefore, square root
of n plus one, plus two
is greater than square root of n plus two,
and if we flip these,
one over square root of
n plus one, plus two,
is less than one over
square root of n plus two.
Therefore, a sub n plus one
is less than a sub n for all n,
and there's our check two,
so our original series converges
by the alternating series test.
