### Notes from Toppers

## 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.