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What can we conclude by applying the πth term divergence test in the series the sum from π equals one to β of three cos π?
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Letβs begin by recalling what the πth term divergence test tells us.
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It tells us that if the limit as π approaches β of π sub π does not exist or if itβs not equal to zero, then the series the sum from π equals one to β of π sub π is divergent.
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We also recall that if the limit is equal to zero, we canβt tell whether the series converges or diverges.
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And so we say that the test fails or itβs inconclusive.
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In this question, we can define π sub π to be equal to three cos π.
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And weβre going to need to evaluate the limit as π approaches β of three cos π.
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Now actually, thereβs not a lot we need to do to evaluate this limit.
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Letβs sketch the graph of three cos π and see if we can recall what it looks like.
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The graph of π¦ equals three cos π₯ is shown.
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We can see that itβs an oscillating function.
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It has peaks and troughs at π¦-values of three and negative three.
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This continues forever.
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And so we say that the limit as π approaches β of three cos π does not exist.
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We cannot pinpoint it as one particular number or even β.
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And by the πth term divergence test, that tells us that the series the sum from π equals one to β of three cos π is divergent.
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It diverges.