Video Transcript
Determine whether the series the
sum from π equals one to β of one over the fifth root of π cubed converges or
diverges.
Now, it may not be immediately
obvious, but if we rewrite the denominator using the general fact that the nth root
of π to the π power equals π to the π over π power, we can write the fifth root
of π cubed as π to the power of three over five. So, we can rewrite our sum as the
sum from π equals one to β of one over π to the power three over five. And we actually recognise this to
be a π-series, which is a series of the form the sum for π equals one to β of one
over π to the π power. And thatβs for any real number
π.
And we actually have a general rule
for determining whether a π-series is convergent or divergent. This is that the π-series the sum
for 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, whatβs the value of π for our
question? Itβs three over five. This is less than or equal to
one. So, by the π test, it
diverges. So, to summarize, by rearrangement
of the denominator, we recognise this to be a π-series. And we applied a general rule for
π-series to reach our conclusion that this series diverges.