### Video Transcript

Determine whether the series the
sum from π equals one to β of one over the square root of π cubed converges
or diverges.

If we start by rewriting this sum,
using the fact that we can write the square root of π as π to the half power, we
can say that this sum is equivalent to the sum from π equals one to β of one
over π cubed raised to the half power. We can then use the fact that π to
the π₯ power, then raised to the π¦ power, equals π to the π₯ multiplied by π¦
power. So we can write our sum as the sum
from π equals one to β of one over π to the three over two power. And then we can recognise this to
be a π-series, which is a series of the form the sum from π equals one to β
of one over π to the π power.

So letβs write out the conditions
for convergence for a π-series. The π-series the sum from π
equals one to β of one over π to the π power is convergent if π is greater
than one and divergent if π is less than or equal to one. So for this series π is equal to
three over two; this is the same as 1.5, which is greater than one. So this series converges.