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

# Question Video: Determine Whether a π-Series Converges or Diverges Mathematics • Higher Education

Determine whether the series β from π = 1 to β of 1/the fifth root of πΒ³ converges or diverges.

01:25

### 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 thatβs 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, whatβs the value of π for our question? Itβs 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.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions