Is n^(3/2) convergent or divergent?
Understand the Problem
The question is asking whether the series or sequence defined by n^(3/2) converges to a finite value or diverges towards infinity as n approaches infinity. To address this, we will analyze the behavior of n^(3/2) as n increases.
Answer
The sequence diverges to infinity.
Answer for screen readers
The series defined by $n^{3/2}$ diverges to infinity as $n$ approaches infinity.
Steps to Solve
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Identify the sequence behavior As $n$ approaches infinity, we need to analyze the expression $n^{3/2}$.
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Evaluate the limit We will find the limit of the sequence as $n$ approaches infinity: $$ \lim_{n \to \infty} n^{3/2} $$
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Determine the limit's value As $n$ becomes very large, $n^{3/2}$ also becomes very large. Thus: $$ \lim_{n \to \infty} n^{3/2} = \infty $$
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Conclusion on convergence Since the limit approaches infinity, we conclude that the sequence diverges.
The series defined by $n^{3/2}$ diverges to infinity as $n$ approaches infinity.
More Information
This means that as we keep increasing $n$, the value of $n^{3/2}$ grows without bound and does not converge to a finite value. This is common in sequences with polynomial growth.
Tips
A common mistake is misinterpreting the limit. It's important to correctly identify that for polynomial sequences like $n^{3/2}$, the values continue to increase without approaching a fixed number.
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