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Determine whether Summation from k equals 2 to infinity StartFraction 1 Over StartRoot k EndRoot minus 1 EndFraction∑k=2[infinity] 1 k−1 converges using the Comparison Test with the comparison series Summation from k equals 1 to infinity StartFraction 1 Over StartRoot x EndRoot EndFraction∑k=1[infinity] 1 k.

Respuesta :

Answer:

Diverges.

Step-by-step explanation:

Given that your sum is

[tex]{\displaystyle \sum\limits_{k=2}^{\infty} \frac{1}{\sqrt{k-1}}[/tex]

notice that

[tex]{\displaystyle \frac{1}{k-1} \leq \frac{1}{\sqrt{k-1}}[/tex]

Since

[tex]{\displaystyle \sum\limits_{k=2}^{\infty} \frac{1}{k-1} \,\,\,\,\,\,\,\text{diverges}[/tex]

using the comparison test

[tex]{\displaystyle \sum\limits_{k=2}^{\infty} \frac{1}{\sqrt{k-1}}[/tex]

diverges.

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