Video: Evaluating the Definite Integral of a Polynomial by Taking the Limit of Riemann Sums

Determine whether the series βˆ‘_(𝑛 = 1)^(∞) (1/βˆšπ‘›Β³) converges or diverges.

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Video Transcript

Determine whether the series the sum from 𝑛 equals one to ∞ of one over the square root of 𝑛 cubed converges or diverges.

If we start by rewriting this sum, using the fact that we can write the square root of π‘Ž as π‘Ž to the half power, we can say that this sum is equivalent to the sum from 𝑛 equals one to ∞ of one over 𝑛 cubed raised to the half power. We can then use the fact that π‘Ž to the π‘₯ power, then raised to the 𝑦 power, equals π‘Ž to the π‘₯ multiplied by 𝑦 power. So we can write our sum as the sum from 𝑛 equals one to ∞ of one over 𝑛 to the three over two power. And then we can recognise this to be 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 conditions for convergence for a 𝑝-series. 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. So for this series 𝑝 is equal to three over two; this is the same as 1.5, which is greater than one. So this series converges.

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