Video Transcript
Determine whether the series the
sum from 𝑛 equals one to ∞ of one over the fifth root of 𝑛 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 𝑎 to the 𝑚 power equals 𝑎 to the 𝑚 over 𝑛 power, we can write the fifth root
of 𝑛 cubed as 𝑛 to the power of three over five. So, we can rewrite our sum as the
sum from 𝑛 equals one to ∞ of one over 𝑛 to the power three over five. And we actually recognise this to
be a 𝑝-series, which is a series of the form the sum for 𝑛 equals one to ∞ of one
over 𝑛 to the 𝑝 power. And that’s for any real number
𝑝.
And we actually have a general rule
for determining whether a 𝑝-series is convergent or divergent. This is that the 𝑝-series the sum
for 𝑛 equals one to ∞ of one over 𝑛 to the 𝑝 power is convergent if 𝑝 is greater
than one and divergent if 𝑝 is less than or equal to one. So, what’s the value of 𝑝 for our
question? It’s three over five. This is less than or equal to
one. So, by the 𝑝 test, it
diverges. So, to summarize, by rearrangement
of the denominator, we recognise this to be a 𝑝-series. And we applied a general rule for
𝑝-series to reach our conclusion that this series diverges.
