Question

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.

Answer 1

Answer:

Check the explanation

Step-by-step explanation:

Kindly check the attached image below to see the step by step explanation to the question above.

Answer 2

Answer:

Diverges.

Step-by-step explanation:

Given that your sum is

${\displaystyle \sum\limits_{k=2}^{\infty} \frac{1}{\sqrt{k-1}}$

notice that

${\displaystyle \frac{1}{k-1} \leq \frac{1}{\sqrt{k-1}}$

Since

${\displaystyle \sum\limits_{k=2}^{\infty} \frac{1}{k-1} \,\,\,\,\,\,\,\text{diverges}$

using the comparison test

${\displaystyle \sum\limits_{k=2}^{\infty} \frac{1}{\sqrt{k-1}}$

diverges.