Question Video: Using Right-Angled Triangle Trigonometry to Find Lengths in Word Problems Mathematics • Higher Education

Video Transcript

Determine whether the series the sum from 𝑛 equals one to ∞ of one over 𝑛 to the power of two-fifths converges or diverges.

We start by recognising that this sum is in the form of a 𝑝-series, which is a series of the form the sum from 𝑛 equals one to ∞ of one over 𝑛 to the 𝑝 power. So let’s write out the condition for convergence for a 𝑝-series. That is the 𝑝-series the sum from 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. And so for this series, 𝑝 equals two-fifths which is less than one. So we can say that this series diverges.