Question

use the integral test to determine whether the series shown below converges or diverges. be sure to check that the conditions of the integral test are satisfied
sum _ { k = 0 } ^ { infty } \frac { 5 } { sqrt { k + 2 } }
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
o a. the series converges because the conditions of the integral test are satisfied and the value of the integral ( int _ { 0 } ^ { infty } \frac { 5 } { sqrt { x + 2 } } d x ) is
(type an exact answer.)
o b. the series diverges because the conditions of the integral test are satisfied and the value of the integral ( int _ { 0 } ^ { infty } \frac { 5 } { sqrt { x + 2 } } d x ) is
(type an exact answer.)
o c. the integral test does not apply to this series.

Explanation:

Step1: Check the conditions of the Integral Test

Let \(f(x)=\frac{5}{\sqrt{x + 2}}\). The function \(f(x)\) is positive for \(x\geq0\) (since the numerator \(5>0\) and the denominator \(\sqrt{x + 2}>0\) when \(x\geq0\)).

To check if it is decreasing, find the derivative. Using the quotient rule, if \(y=\frac{5}{(x + 2)^{\frac{1}{2}}}=5(x + 2)^{-\frac{1}{2}}\), then \(y^\prime=5\times(-\frac{1}{2})(x + 2)^{-\frac{3}{2}}=\frac{-5}{2(x + 2)^{\frac{3}{2}}}<0\) for \(x\geq0\). So the function is continuous (as a rational - root function) and decreasing for \(x\geq0\). The conditions of the Integral Test are satisfied.

Step2: Evaluate the improper integral \(\int_{0}^{\infty}\frac{5}{\sqrt{x + 2}}dx\)

First, find the antiderivative. Let \(u=x + 2\), then \(du=dx\).

\(\int\frac{5}{\sqrt{x + 2}}dx=5\int u^{-\frac{1}{2}}du\)

Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), we have \(5\int u^{-\frac{1}{2}}du=5\times\frac{u^{\frac{1}{2}}}{\frac{1}{2}}+C = 10\sqrt{u}+C=10\sqrt{x + 2}+C\)

Now, evaluate the improper integral:

\(\int_{0}^{\infty}\frac{5}{\sqrt{x + 2}}dx=\lim_{b
ightarrow\infty}\int_{0}^{b}\frac{5}{\sqrt{x + 2}}dx=\lim_{b
ightarrow\infty}[10\sqrt{x + 2}]_{0}^{b}\)

\(=\lim_{b
ightarrow\infty}(10\sqrt{b + 2}-10\sqrt{0 + 2})\)

As \(b
ightarrow\infty\), \(\sqrt{b+2}
ightarrow\infty\), so \(\lim_{b
ightarrow\infty}(10\sqrt{b + 2}-10\sqrt{2})=\infty\)

Answer:

B. The series diverges because the conditions of the Integral Test are satisfied and the value of the integral \(\int_{0}^{\infty}\frac{5}{\sqrt{x + 2}}dx\) is \(\infty\)