This PPT includes Convergence and Divergence of Improper Integrals 1)Limit comparison test 2)Direct comparison test

1. Gandhinagar Institute of Technology(012)
Active Learning Assignment
Subject- Calculus(2110014)
Topic- Convergence and Divergence of Improper Integrals
Branch-Computer Engineering C : C-2
Prepared By- Somai Rohankumar J.(170120107169)
Guided By- Prof. Mansi Vaishnani

2. Agenda
• Direct Comparison Test
• Example
• Limit Comparison Test
• Example

3. Direct Comparison Test
(a) If f(x) and g(x) are two continuous functions for x ≥ a such that
0 ≤ f(x) ≤ g(x) then,
𝑎
∞
𝑓(𝑥) 𝑑𝑥 converges, if 𝑎
∞
𝑔(𝑥) 𝑑𝑥 converges and
𝑎
∞
𝑓 𝑥 𝑑𝑥 ≤ 𝑎
∞
𝑔(𝑥) 𝑑𝑥.

4. Direct Comparison Test (contd.)
(b) If f(x) and g(x) are two continuous functions for x ≥ a such that
f(x) ≥ g(x) then,
𝑎
∞
𝑓(𝑥) 𝑑𝑥 diverges , if 𝑎
∞
𝑔(𝑥) 𝑑𝑥 diverges.

5. Example
• Test the convergence of the improper integral 1
∞ 𝑐𝑜𝑠𝑥
𝑥2 𝑑𝑥 .
• Solution

6. Example (contd.)

7. Example (contd.)

8. Limit Comparison Test
• If f(x) and g(x) are two continuous functions for x ≥ a such
that
f(x) > 0 , g(x) > 0 and lim
𝑥→∞
𝑓(𝑥)
𝑔(𝑥)
= l , 0 < l < ∞ then
𝑎
∞
𝑓 𝑥 𝑑𝑥 and 𝑎
∞
𝑔 𝑥 𝑑𝑥 both converge or
diverge simultaneously.

9. Example
• Test the convergence of the improper integral 4
∞ 3𝑥+5
𝑥4+7
𝑑𝑥 .
• Solution

10. Example (contd.)

11. Example (contd.)

12. Example (contd.)

13. References
• Calculus Textbook Published by Mc Graw Hill Eduation
Written by Ravish R Singh
& Mukul Bhatt