Use the Limit Comparison Test to determine whether the series converges. [infinity]∑ from k = 1 StartFraction 8/k StartRoot k + 7 EndRoot EndFraction

The infinite series indeed converges.

Step-by-step explanation:

The limit comparison test for infinite series of positive terms compares the convergence of an infinite sequence (where all terms are greater than zero) to that of a similar-looking and better-known sequence (for example, a power series.)

For example, assume that it is known whether converges or not. Compute the following limit to study whether converges:

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Let denote each term of this infinite Rewrite the infinite sequence in this question:

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Compare that to the power series where . Note that this

Verify that all terms of are indeed greater than zero. Apply the limit comparison test:

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Note, that both the square root function and fractions are continuous over all real numbers. Therefore, it is possible to move the limit inside these two functions. That is:

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Because the limit of this ratio is a finite positive number, it can be concluded that the convergence of and are the same. Because the power series converges, (by the limit comparison test) the infinite series should also converge.

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step-by-step explanation: