# Section 5.6 Calculus 2

Ratio and Root Tests

## 5.6 Ratio and Root Tests

### 5.6.1 Ratio Test

• The Ratio Test states that the series $$\sum_{n=N}^\infty a_n$$ converges when $$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|<1$$ and diverges when $$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|>1$$.
• Example Show that $$\sum_{n=0}^\infty\frac{3^n+1}{4^n}$$ converges using the Ratio Test. Then give its value.
• Example Does $$\sum_{k=3}^\infty\frac{(2k)!}{3(k!)^2}$$ converge or diverge?
• Another test must be used when $$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|=1$$.
• Example Show that the divergent series $$\sum_{n=1}^\infty\frac{1}{n}$$ and the convergent series $$\sum_{n=1}^\infty\frac{1}{n^2}$$ both satisfy $$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|=1$$.

### 5.6.2 Root Test

• The Root Test states that the series $$\sum_{n=N}^\infty a_n$$ converges when $$\lim_{n\to\infty}\sqrt[n]{|a_n|}<1$$ and diverges when $$\lim_{n\to\infty}\sqrt[n]{|a_n|}>1$$.
• Example Show that $$\sum_{n=0}^\infty\frac{5^n}{2^{3n}}$$ converges using the Root Test. Then give its value.
• Example Does $$\sum_{m=3}^\infty\frac{m^{10}}{(-3)^m}$$ converge or diverge?
• Another test must be used when $$\lim_{n\to\infty}\sqrt[n]{|a_n|}=1$$.

### Review Exercises

1. Does $$\sum_{k=1}^\infty\frac{k^2+4}{(k+2)!}$$ converge or diverge?
2. Does $$\sum_{n=0}^\infty\frac{(2n)!}{n+3}$$ converge or diverge?
3. Does $$\sum_{m=2}^\infty\frac{5^m}{m!}$$ converge or diverge?
4. Does $$\sum_{n=0}^\infty(-1)^n\frac{n!}{2^n(n+2)!}$$ converge or diverge?
5. Does $$\sum_{p=0}^\infty\frac{3^p}{(p+7)^p}$$ converge or diverge?
6. Does $$\sum_{n=9}^\infty(1+\frac{2}{n})^{n^2}$$ converge or diverge? (Hint: $$e^x=\lim_{n\to\infty}(1+\frac{x}{n})^n$$.)
7. Does $$\sum_{j=3}^\infty(-3)^j\frac{1}{j4^j}$$ converge or diverge?
8. Does $$\sum_{n=1}^\infty\left(\frac{1-4n^2}{(n+1)(3n+1)}\right)^{n+3}$$ converge or diverge?
9. Does $$\sum_{m=4}^\infty(-1)^{m+1}\frac{me^{-m}}{(2m+1)\ln(m+1)}$$ converge or diverge?
10. Does $$\sum_{n=1}^\infty\frac{(n-1)!}{10^n}$$ converge or diverge?
11. Does $$\sum_{k=3}^\infty(1-\frac{1}{k})^{k^2}$$ converge or diverge
12. Does $$\sum_{m=2}^\infty\frac{1}{m^2}$$ converge or diverge?

Solutions

### Textbook Reference

• University Calculus: Early Transcendentals (3rd Ed)
• 9.5