Question

Use the Limit Comparison Test to determine whether the series converges.

[infinity]∑ from k = 1 StartFraction 8/k StartRoot k + 7 EndRoot EndFraction

Answer 1

The infinite series
`\displaystyle \sum\limits_(k = 1)^(\infty) (8/k)/(√(k + 7))` indeed converges.

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
`\displaystyle \sum\limits_(k = 1)^(\infty) b_k` converges or not. Compute the following limit to study whether
`\displaystyle \sum\limits_(k = 1)^(\infty) a_k` converges:

`\displaystyle \lim\limits_(k \to \infty) (a_k)/(b_k)\; \begin{tabular}{l}\\ $\leftarrow$ Series whose convergence is known\end{tabular}`.

• If that limit is a finite positive number, then the convergence of the these two series are supposed to be the same.
• If that limit is equal to zero while
  `a_k` converges, then
  `b_k` is supposed to converge, as well.
• If that limit approaches infinity while
  `a_k` does not converge, then
  `b_k` won't converge, either.

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

`\begin{aligned}a_k &= (8/k)/(√(k + 7))\\ &= (8)/(k\cdot √(k + 7)) = (8)/(√(k^2\, (k + 7))) = (8)/(√(k^3 + 7\, k^2)) \end{aligned}`.

Compare that to the power series
`\displaystyle \sum\limits_(k = 1)^(\infty) b_k` where
`\displaystyle b_k = (1)/(√(k^3)) = (1)/(k^(3/2)) = k^(-3/2)`. Note that this

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

`\begin{aligned}& \lim\limits_(k \to \infty) (a_k)/(b_k)\; \begin{tabular}{l}\\ $\leftarrow$ Series whose convergence is known\end{tabular}\\ &= \lim\limits_(k \to \infty) \frac{\displaystyle (8)/(√(k^3 + 7\, k^2))}{\displaystyle \frac{1}{{√(k^3)}}}\\ &= 8\left(\lim\limits_(k \to \infty) \sqrt{(k^3)/(k^3 + 7\, k^2)}\right) = 8\left(\lim\limits_(k \to \infty) \sqrt{(1)/(\displaystyle 1 + (7/k))}\right)\end{aligned}`.

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:

`\begin{aligned}& \lim\limits_(k \to \infty) (a_k)/(b_k)\\ &= \cdots \\ &= 8\left(\lim\limits_(k \to \infty) \sqrt{(1)/(\displaystyle 1 + (7/k))}\right)\\ &= 8\left(\sqrt{(1)/(\displaystyle 1 + \lim\limits_(k \to \infty) (7/k))}\right) \\ &= 8\left(\sqrt{(1)/(1 + 0)}\right) \\ &= 8 \end{aligned}`.

Because the limit of this ratio is a finite positive number, it can be concluded that the convergence of
`\displaystyle a_k &= (8/k)/(√(k + 7))` and
`\displaystyle b_k = (1)/(√(k^3))` are the same. Because the power series
`\displaystyle \sum\limits_(k = 1)^(\infty) b_k` converges, (by the limit comparison test) the infinite series
`\displaystyle \sum\limits_(k = 1)^(\infty) a_k` should also converge.