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Use the π-series test to determine whether the series the sum from π equals one to β of one divided by four π is divergent or convergent.
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The question gives us an infinite series.
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It wants us to determine whether this series is divergent or convergent by using a π-series test.
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Letβs start by recalling what we mean by the π-series test.
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We call the sum from π equals one to β of one divided by π to the πth power a π-series.
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And we know that this series is convergent when π is greater than one and divergent when π is less than or equal to one.
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A π-series test means to compare our series to a π-series.
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We can then determine the convergence or divergence by using this test.
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So, we want to compare the series given to us in the question, thatβs the sum from π equals one to β of one divided by four π, with a π-series.
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To start, we can notice the π in our denominator can be rewritten as π to the first power.
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We can now see our series is almost in the form of a π-series.
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We just have this constant factor of four in our denominator.
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But this is a constant factor, so we can take this outside of our sum.
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This wonβt change the convergence or divergence of our series.
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In other words, we can rewrite our series as one-quarter times the sum from π equals one to β of one divided by π to the first power.
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We can now see this is a constant multiple of a π-series where π is equal to one.
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And we know when π is equal to one, our π-series will be divergent.
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In fact, when π is equal to one, we call this series the harmonic series.
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So, the series given to us in the question is one-quarter times our divergent series.
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This means the series given to us in the question is also divergent.
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Therefore, by using the π-series test, we were able to show the sum from π equals one to β of one divided by four π is divergent.