# Section 5.5 Calculus 2

The Integral Test

## 5.5 The Integral Test

### 5.5.1 Improper Integrals

• 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 $$\int_0^8 z^{-1/3}\,dz$$.

### 5.5.2 The Integral Test

• 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 $$\sum_{n=1}^\infty\frac{1}{n^3}\not=\int_1^\infty\frac{1}{x^3}\,dx$$.

### 5.5.3 The $$p$$-Series Test

• 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?

### Review Exercises

1. Does $$\int_2^\infty\frac{32}{x^3}\,dx$$ converge or diverge? If it converges, what is its value?
2. Does $$\int_0^\infty\frac{2y}{y^2+3}\,dy$$ converge or diverge? If it converges, what is its value?
3. Does $$\int_e^\infty\frac{1}{\ln(x^x)}\,dx$$ converge or diverge? If it converges, what is its value?
4. 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.
5. Does $$\sum_{n=0}^\infty\frac{2n}{n^2+3}$$ converge or diverge?
6. Does $$\sum_{n=3}^\infty\frac{4}{n(\ln n)^3}$$ converge or diverge?
7. Does $$\sum_{n=-2}^\infty\frac{1}{e^n}$$ converge or diverge?
8. Show that $$\int_1^\infty\frac{1}{x^2}\,dx \not= \sum_{n=1}^\infty\frac{1}{n^2}$$, even though they both converge.
9. Does $$\sum_{k=100}^\infty\frac{5}{\sqrt[7]{k^6}}$$ converge or diverge?
10. Does $$\sum_{n=5}^\infty\frac{1}{n^2-8n+16}$$ converge or diverge?
11. 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 $$\lim_{u\to\infty}\tan^\leftarrow u=\frac{\pi}{2}$$.)
12. Does $$\sum_{m=0}^\infty\frac{2m}{(m^2+1)^2}$$ converge or diverge?
13. Does $$\sum_{n=2}^\infty\frac{1}{\sqrt{n-1}}$$ converge or diverge?

Solutions 1-6

Solutions 7-13

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 8.7, 9.3