# Section 5.7 Calculus 2

Comparison Tests

## 5.7 Comparison Tests

### 5.7.1 Direct Comparison Test

• Following is a list of sequence formulas ordered from larger to smaller (for sufficiently large $$n$$).
• $$n^n$$
• $$n!$$
• $$b^n$$ where $$b>1$$ (such as $$2^n,e^n,10^n$$…)
• $$n^p$$ where $$p>0$$ (such as $$\sqrt{n},n,n^4$$…)
• $$\log_b n$$ where $$b>1$$ (such as $$\log_{10}(n),\ln(n),\log_2(n)$$…)
• any positive constant
• Example Show that $$\frac{m^3+7}{4^m+5}\leq 2(\frac{1}{2})^m$$ for sufficiently large values of $$m$$.
• Suppose $$\sum_{n=N}^\infty a_n$$ is a series with non-negative terms.
• If there exists a convergent series $$\sum_{n=M}^\infty b_n$$ with non-negative terms where $$a_n\leq b_n$$ for sufficiently large $$n$$, then $$\sum_{n=N}^\infty a_n$$ converges as well.
• If there exists a divergent series $$\sum_{n=M}^\infty b_n$$ with non-negative terms where $$a_n\geq b_n$$ for sufficiently large $$n$$, then $$\sum_{n=N}^\infty a_n$$ diverges as well.
• Example Show that $$\sum_{m=0}^\infty \frac{m^3+7}{4^m+5}$$ converges by comparing with the series $$\sum_{m=0}^\infty 2(\frac{1}{2})^m$$.
• Example Does $$\sum_{n=1}^\infty\frac{2}{n^{1/3}+5}$$ converge or diverge?
• Example Does $$\sum_{k=3}^\infty\frac{e^k}{e^{2k}-1}$$ converge or diverge?
• Example Does $$\sum_{m=2}^\infty(m\ln m)^{-1/2}$$ converge or diverge?

### 5.7.2 Limit Comparison Test

• Suppose $$\sum_{n=N}^\infty a_n$$ is a series with non-negative terms. If there exists a series $$\sum_{n=M}^\infty b_n$$ with non-negative terms where $$0<\lim_{n\to\infty}\frac{a_n}{b_n}<\infty$$, then either both series converge or both series diverge.
• Example Does $$\sum_{n=1}^\infty\frac{2}{n^{1/3}+5}$$ converge or diverge?
• Example Does $$\sum_{i=0}^\infty\frac{i^2+3i+7}{3i^4+2i^2+5}$$ converge or diverge?
• Example Does $$\sum_{n=42}^\infty\frac{2^n+5^n}{3^n+4^n}$$ converge or diverge?

### Review Exercises

1. Does $$\sum_{n=0}^\infty\sqrt{\frac{n}{n^4+7}}$$ converge or diverge?
2. Does $$\sum_{n=3}^\infty\frac{4}{n^{0.8}-1}$$ converge or diverge?
3. Does $$\sum_{j=2}^\infty\frac{e^j}{e^{2j}+1}$$ converge or diverge?
4. Does $$\sum_{k=10}^\infty\frac{\sin^2(k)}{k^3}$$ converge or diverge?
5. Does $$\sum_{m=4}^\infty\frac{1}{\ln m}$$ converge or diverge?
6. Does $$\sum_{n=4}^\infty\frac{5}{2n+3}$$ converge or diverge?
7. Does $$\sum_{m=1}^\infty\frac{1}{1+2+\dots+(m-1)+m}$$ converge or diverge? (Hint: show that $$\frac{1}{1+2+\dots+(m-1)+m} = \frac{2}{(1+m)+(2+m-1)+\dots+(m-1+2)+(m+1)}$$.)
8. Does $$\sum_{m=0}^\infty\frac{2m}{(m^2+1)^2}$$ converge or diverge?
9. Does $$\sum_{n=1}^\infty\sqrt{\frac{n+1}{n^2+3}}$$ converge or diverge?

Solutions

## Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 9.4