Question Video: Determine Whether a 𝑝-Series Converges or Diverges

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.