Section 5.5 Calculus 2 - Professor Clontz

5.5 The Integral Test

If \(f(x)\geq 0\), the improper integral \(\int_a^\infty f(x)\,dx=\lim_{b\to\infty}\int_a^b f(x)\,dx\) represents the area under the curve \(y=f(x)\) from \(x=a\) out to \(\infty\). If the limit exists, then the improper integral converges; otherwise it diverges.
Example Does \(\int_1^\infty\frac{1}{x^2}\,dx\) converge or diverge? If it converges, what is its value?
Example Does \(\int_4^\infty\frac{1}{2\sqrt y}\,dy\) converge or diverge? If it converges, what is its value?
When an integrand is undefined at a bound of integration, then the integral is also called improper and is evaluated with a limit.
Example Find the value of \(\int_0^8 z^{-1/3}\,dz\).

If \(a_n=f(n)\) where \(f(x)\) is a continuous, positive, decreasing function for sufficiently large values of \(x\), then the series \(\sum_{n=N}^\infty a_n\) and improper integral \(\int_a^\infty f(x)\,dx\) either both converge, or both diverge.
Example Does \(\sum_{n=4}^\infty\frac{4n+4}{n^2+2n+1}\) converge or diverge?
Example Does \(\sum_{k=1}^\infty\frac{k}{e^{k^2}}\) converge or diverge?
Even when they both converge, the values of the series \(\sum_{n=N}^\infty a_n\) and improper integral \(\int_N^\infty f(x)\,dx\) usually differ.
Example Show that \(\sum_{n=1}^\infty\frac{1}{n^3}\not=\int_1^\infty\frac{1}{x^3}\,dx\).

The \(p\)-Series Test states that the series \(\sum_{n=1}^\infty\frac{1}{n^p}\) converges when \(p>1\), and diverges when \(p\leq 1\).
Example Does \(\sum_{m=2}^\infty\frac{3}{\sqrt[10]{m^4}}\) converge or diverge?
Example Does \(\sum_{j=0}^\infty\frac{1}{j^2+2j+1}\) converge or diverge?

Does \(\int_2^\infty\frac{32}{x^3}\,dx\) converge or diverge? If it converges, what is its value?
Does \(\int_0^\infty\frac{2y}{y^2+3}\,dy\) converge or diverge? If it converges, what is its value?
Does \(\int_e^\infty\frac{1}{\ln(x^x)}\,dx\) converge or diverge? If it converges, what is its value?
Show that \(\int_1^\infty\frac{1}{x^2}\,dx+1=\int_0^1\frac{1}{\sqrt y}\,dy\). Then draw a sketch involving areas illustrating why they are equal.
Does \(\sum_{n=0}^\infty\frac{2n}{n^2+3}\) converge or diverge?
Does \(\sum_{n=3}^\infty\frac{4}{n(\ln n)^3}\) converge or diverge?
Does \(\sum_{n=-2}^\infty\frac{1}{e^n}\) converge or diverge?
Show that \( \int_1^\infty\frac{1}{x^2}\,dx \not= \sum_{n=1}^\infty\frac{1}{n^2} \), even though they both converge.
Does \(\sum_{k=100}^\infty\frac{5}{\sqrt[7]{k^6}}\) converge or diverge?
Does \(\sum_{n=5}^\infty\frac{1}{n^2-8n+16}\) converge or diverge?
Does \(\sum_{n=-1}^\infty\frac{e^n}{1+e^{2n}}\) converge or diverge? (Hint: \(\int\frac{1}{1+u^2}\,du=\tan^\leftarrow u+C\) and \(\lim_{u\to\infty}\tan^\leftarrow u=\frac{\pi}{2}\).)
Does \(\sum_{m=0}^\infty\frac{2m}{(m^2+1)^2}\) converge or diverge?
Does \(\sum_{n=2}^\infty\frac{1}{\sqrt{n-1}}\) converge or diverge?