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

Check the explanation

Explanation:

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

Answer 2

Diverges.

Explanation:

Given that your sum is

`{\displaystyle \sum\limits_(k=2)^(\infty) (1)/(√(k-1))`

notice that

`{\displaystyle (1)/(k-1) \leq (1)/(√(k-1))`

Since

`{\displaystyle \sum\limits_(k=2)^(\infty) (1)/(k-1) \,\,\,\,\,\,\,\text{diverges}`

using the comparison test

`{\displaystyle \sum\limits_(k=2)^(\infty) (1)/(√(k-1))`

diverges.