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Determine whether the series the sum from π equals one to β of one over π to the power of two-fifths converges or diverges.
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We start by recognising that this sum is in the form of 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.
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So letβs write out the condition for convergence for a π-series.
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That is 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.
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And so for this series, π equals two-fifths which is less than one.
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So we can say that this series diverges.