Textbook Section 11.3 Question 11 Solutions
Question
Determine whether the series is convergent or divergent.
We use the Integral Test here which is:
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If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is convergent.
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If \(\int_1^\infty f(x) \, dx\) convergent, then \(\sum_{n=1}^{\infty} a_n\) is divergent.
Then you basically take the \(\lim_{m \to \infty} \int_1^m f(x) \, dx\) which in this case is \(\frac{1}{n^{\sqrt{2}}}\).
You proceed to take the integral with Power Rule and then finally taking the limit.
Since the limit exists, it converges which is the final answer.
The power-P rule of the Integral Test can also be applied here!
Solutions
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