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

Determine whether the series β_(π = 1)^(β) (1/βπΒ³) converges or diverges.

01:15

### Video Transcript

Determine whether the series the sum from π equals one to β of one over the square root of π cubed converges or diverges.

If we start by rewriting this sum, using the fact that we can write the square root of π as π to the half power, we can say that this sum is equivalent to the sum from π equals one to β of one over π cubed raised to the half power. We can then use the fact that π to the π₯ power, then raised to the π¦ power, equals π to the π₯ multiplied by π¦ power. So we can write our sum as the sum from π equals one to β of one over π to the three over two power. And then we can recognise this to be a π-series, which is a series of the form the sum from π equals one to β of one over π to the π power.

So letβs write out the conditions for convergence for a π-series. The π-series the sum from 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 for this series π is equal to three over two; this is the same as 1.5, which is greater than one. So this series converges.