Which of the following statements are​ true?i.If a_{n} and f(n) satisfy the requirements of the integral test, then \sum_{n=1}^{\infty} a_n = \int_1^{\infty} f(x)dx.ii.

preprekomW 2021-06-11 Answered

Which of the following statements are​ true?
i.If \(a_{n}\ \text{and}\ f(n)\) satisfy the requirements of the integral test, then \(\sum_{n=1}^{\infty} a_n = \int_1^{\infty} f(x)dx\)
ii. The series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\ \text{converges if}\ p > 1\ \text{and diverges if}\ p \leq 1\).
iii. The integral test does not apply to divergent sequences.

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Expert Answer

Nathanael Webber
Answered 2021-06-12 Author has 12162 answers

1)If \(a_n\) and f(n) satisty the requirements of the integral test, then \(\sum_{n=1}^{\infty} a_n = \int_1^{\infty} f(x)dx\)
2)The series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\ \text{converges if}\ p > 1\ \text{and diverges if}\ p \leq 1\).
3) The integral test can be applied to both covergent and divergent testedherefore E.Only statements 1 and 2 are true.

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