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Determine whether the series the sum from π equals one to β of one divided by π times the square root of π cubed converges or diverges.
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Weβre given a series, and weβre asked to determine whether this series converges or diverges.
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And thereβs a lot of different ways of dealing with this.
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For example, we need to check that our terms are getting closer and closer to zero.
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This is called the πth term divergence test.
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We need to check the limit as π approaches β our summand approaches zero.
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And in this case, this is true.
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We can see this because our numerator remains constant; however, our denominator is growing without bound.
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And at this point, thereβs a few different things we could try.
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We could start looking at our partial sums; however, in this case, thereβs actually an easier method.
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If we look at our summand, we can see we can rearrange this by using our laws of exponents.
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And in this case, we will be able to rewrite this as a π-series.
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So, weβll use the π-series test to determine the convergence or divergence of this series.
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So, letβs start by recalling the π-series test.
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This tells the sum from π equals one to β of one divided by π to the power of π is convergent if π is greater than one and divergent if π is less than or equal to one.
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To use this, weβre going to need to use our laws of exponents to rewrite our series.
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First, recall taking the square root of a number is the same as raising that number to the power of one-half.
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So, the square root of π cubed is π cubed raised to the power of one-half.
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But we can then simplify this even further.
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π cubed all raised to the power of one-half is equal to π to the power of three times one-half, which is π to the power of three over two.
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So, we were able to rewrite our series as the sum from π equals one to β of one divided by π times π to the power of three over two.
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But we can simplify this even further by using our laws of exponents.
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By writing π as π to the first power, we can multiply π to the first power by π to the power of three over two by adding our exponents.
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Doing this, we get the sum from π equals one to β of one divided by π to the power of one plus three over two.
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And, of course, we can simplify this.
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One plus three over two is equal to five over two.
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So, we were able to rewrite the power series given to us in the question as the sum from π equals one to β of one divided by π to the power of five over two.
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And we can see this is exactly equal to a π-series, where the value of π is equal to five over two and five over two is greater than one.
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So, by our π-series test, this means that our series must be convergent.
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Therefore, by using our laws of exponents, we were able to show the sum from π equals one to β of one divided by π times the square root of π cubed is a π-series where π is equal to five over two.
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And then, by using the π-series test, we can conclude that this means our series must be convergent.