Here are the important subtopics within the topic of “Infinite Series” for the JEE exam, along with references to NCERT books for 11th and 12th classes:

1. Convergence and Divergence:

• NCERT Reference: NCERT Class 11, Chapter 9 (Sequences and Series): Sections 9.1, 9.2, 9.3, 9.4, 9.5.
• Series Convergence Tests: Comparison Test, Limit Comparison Test, Ratio Test, Root Test, Integral Test, Cauchy’s Root Test, Alternating Series Test, and more.
• Series Divergence Tests: Series with positive terms, Series with non-positive terms, Oscillating Series, and Harmonic Series.

2. Telescoping Series:

• NCERT Reference: NCERT Class 11, Chapter 9 (Sequences and Series): Section 9.3.
• Definition and understanding the concept of telescoping series.
• Solving telescoping sums and determining their convergence.

3. Power Series:

• NCERT Reference: NCERT Class 12, Chapter 8 (Applications of Derivatives): Section 8.5, NCERT Class 11, Chapter 9 (Sequences and Series): Section 9.8.
• Definition of a power series and interval of convergence.
• Finding the interval of convergence using Ratio Test, Root Test, or other methods.
• Power series representations of elementary functions (exponential, logarithmic, trigonometric, and inverse trigonometric functions).

4. Differentiation and Integration of Power Series:

• NCERT Reference: NCERT Class 12, Chapter 8 (Applications of Derivatives): Section 8.5.
• Differentiating and integrating power series term-by-term (within the interval of convergence).

5. Taylor Series:

• NCERT Reference: NCERT Class 12, Chapter 8 (Applications of Derivatives): Section 8.5.
• Taylor’s Series Expansion: Definition and the concept of Taylor’s polynomial approximations.
• Finding Taylor series expansions for various functions.
• Applications of Taylor series, including approximating functions, finding limits, and solving differential equations.

6. Series of Functions:

• NCERT Reference: NCERT Class 12, Chapter 8 (Applications of Derivatives): Section 8.4.
• Uniform convergence and pointwise convergence of series of functions.
• Tests for uniform convergence, such as the Weierstrass M-Test and Abel’s Test.

7. Fourier Series:

• NCERT Reference: NCERT Class 12, Chapter 12 (Application of Integrals): Section 12.1.
• Understanding the basics of Fourier series, including periodic functions, complex form of Fourier series, and determining coefficients.

8. Applications:

• NCERT Reference: NCERT Class 11, Chapter 9 (Sequences and Series): Applications across several chapters.
• Applying infinite series to solve problems in various fields such as calculus, physics, and engineering.

9. Improper Integrals:

• NCERT Reference: NCERT Class 12, Chapter 7 (Integrals): Section 7.11.
• Evaluating improper integrals using techniques involving infinite series.

10. Complex Analysis:

• Basics of complex numbers, complex functions, and their relationship to infinite series.

11. Calculus of Variations:

• Applying infinite series to optimize functionals and solve problems in the calculus of variations.

12. Generating Functions:

• Definition of generating functions and understanding how they relate to sequences and series.

Remember to cover these subtopics with regular practice and a thorough understanding of the concepts to excel in the Infinite Series section of the JEE exam.