Use Limit Comparison Test or Direct Comparison Test?

💬Question: We start with the question that when should I use for improper integral?

• direct comparison test ? or
• limit comparison test ?

🗣Answer: When you can’t do a direct comparison, use the limit comparison.

📝Definition: click the footnote to review the concept if needed.

• direct comparison test¹
• limit comparison test²
• improper integral³

🗃Example: Investigate the convergence of $\displaystyle \int_1^{\infty}\frac{1-e^{-x}}{x}dx$

✍Step by Step Reasoning:

• By the improper integral³, we know that the integrand suggests a comparison of
  • $\displaystyle \textcolor{purple}{f(x)=\frac{1-e^{-x}}{x}}$ and
  • $\displaystyle \textcolor{pink}{g(x)=\frac{1}{x}}$
  • because $\displaystyle \textcolor{purple}{\frac{1-e^{-x}}{x}} < \textcolor{pink}{\frac{1}{x}}$ in $\displaystyle [1,\infty)$.
• However, we cannot use the Direct Comparison Test¹.
  • from its definition we know that when $f(x) \leq g(x)$, the antecedents are
    • $\color{pink}{\displaystyle \int _a^{\infty } g(x) \, dx}$ converges, or
    • $\color{purple}{\displaystyle \int _a^{\infty } f(x) \, dx}$ diverges
  • however, in this case, the $\color{pink}{\displaystyle \int _a^{\infty } g(x) \, dx}$ diverges… This doesn’t obey the rule.
  • Meaning, the rule of direct comparison suggests that $\color{purple}{\displaystyle \int _a^{\infty } f(x) \, dx}$ diverges is antecedent (因) and $\color{pink}{\displaystyle \int _a^{\infty } g(x) \, dx}$ diverges is consequent (果).
  • Now it’s the opposite.
• Therefore, we should use the Limit Comparison Test².
  • using the Limit Comparison Test we find that
    • $\displaystyle \lim_{x\to\infty}\frac{f(x)}{g(x)}=\lim_{x\to\infty}\left(\frac{1-e^{-x}}{x}\right)\left(\frac{x}{1}\right)=\lim_{x\to\infty}(1-e^{-x})=1$
  • which is a positive finite limit.
• Result.
  • Therefore, $\displaystyle \int_1^{\infty}\frac{1-e^{-x}}{x}dx$ diverges because $\displaystyle \int_1^{\infty}\frac{1}{x}dx$ diverges.

📜Key Takeaway:

• Use limit comparison test✅ when you can’t use direct comparison test🙈.
• Identify what is antecedent and what is consequent in the mathematical definition.

🔗Reference: Thomas Calculus > Chapter 8 Techniques of Integration > 8.8 Improper Integrals