Notes from Toppers
Sequences and Series
1. Sequences

Arithmetic progression (AP):
 Sum of n terms of an AP (NCERT Class 11, Chapter 9, Section 9.1):
 Formula: $$S_n = n/2(2a + (n  1)d),$$ where S_{n} is the sum of the first n terms, a is the first term, and d is the common difference.
 NCERT Examples: Solved Examples 9.1 (a), (b), (c)
 nth term of an AP (NCERT Class 11, Chapter 9, Section 9.1):
 Formula: $$T_n = a + (n  1)d,$$ where T_{n} is the nth term of the AP.
 NCERT Examples: Solved Examples 9.1 (d), (e), (f)
 Sum of n terms of an AP (NCERT Class 11, Chapter 9, Section 9.1):

Geometric progression (GP):
 Sum of n terms of a GP (NCERT Class 11, Chapter 9, Section 9.2):
 Formula: $$S_n = \frac{a(r^n  1)}{r1},$$ where S_{n} is the sum of the first n terms, a is the first term, and r is the common ratio.
 NCERT Examples: Solved Examples 9.2 (a), (b), (c)
 nth term of a GP (NCERT Class 11, Chapter 9, Section 9.2):
 Formula: $$T_n = ar^{n1},$$ where T_{n} is the nth term of the GP.
 NCERT Examples: Solved Examples 9.2 (d), (e), (f)
 Sum of n terms of a GP (NCERT Class 11, Chapter 9, Section 9.2):

Harmonic progression (HP):
 Sum of n terms of a HP (NCERT Class 11, Chapter 9, Section 9.3):
 Formula: $$S_n = n(H_1 + H_n)/2,$$ where S_{n} is the sum of the first n terms, H_{1} is the first term, and H_{n} is the nth term.
 NCERT Examples: Solved Examples 9.3 (a), (b), (c)
 nth term of a HP (NCERT Class 11, Chapter 9, Section 9.3):
 Formula: $$H_n = \frac{n}{a + (n  1)d},$$ where H_{n} is the nth term of the HP, a is the first term, and d is the common difference.
 NCERT Examples: Solved Examples 9.3 (d), (e), (f)
 Sum of n terms of a HP (NCERT Class 11, Chapter 9, Section 9.3):
2. Summation of Series

Series involving arithmetic progressions (NCERT Class 12, Chapter 9, Section 9.2):
 Formula: $$S = \frac{n}{2}(2a + (n  1)d),$$ where S is the sum of the series, a is the first term, d is the common difference, and n = infinity (for an infinite series).
 NCERT Examples: Solved Examples 9.2 (a), (b), (c), (d), (e)

Series involving geometric progressions (NCERT Class 12, Chapter 9, Section 9.3):
 Formula: $$S = \frac{a}{1  r},$$ where S is the sum of the series, a is the first term, and r is the common ratio with r < 1 (for an infinite series).
 NCERT Examples: Solved Examples 9.3 (a), (b), (c), (d), (e)

Series involving harmonic progressions:
 Use the series formula for the harmonic progression based on the form 1/a, 1/(a + d), 1/(a + 2d), …, 1/[a + (n  1)d], or use appropriate summation techniques.

Telescoping series (NCERT Class 12, Chapter 9, Section 9.5):
 Formula: If $$T_n = a_{n+1}  a_n,$$ then $$S = T_1 + T_2 + T_3 + … + T_n = a_{n+1}  a_1$$
 NCERT Examples: Solved Examples 9.5 (a), (b), (c), (d)

Partial sums (NCERT Class 12, Chapter 9, Section 9.1):
 For a given series, the nth partial sum is the sum of the first n terms of the series.
3. Convergence and Divergence of Series

Cauchy’s criterion for convergence (NCERT Class 12, Chapter 9, Section 9.7):
 A series converges if and only if the limit of its nth partial sums is a finite number.
 NCERT Examples: Solved Examples 9.7 (a), (b), (c), (d), (e)

D’Alembert’s ratio test (NCERT Class 12, Chapter 9, Section 9.8):
 Let (L = \lim_{n \to \infty} \left\frac{a_{n+1}}{a_n}\right).
 If (L < 1, the series converges.
 If (L > 1,) or if (L) does not exist or is infinite, the series diverges.
 NCERT Examples: Solved Examples 9.8 (a), (b), (c), (d), (e), (f)
 Let (L = \lim_{n \to \infty} \left\frac{a_{n+1}}{a_n}\right).

Raabe’s test:
 Let (L = \lim_{n \to \infty} n\left\frac{a_{n+1}}{a_n}  1\right).
 If (L < 1, the series converges.
 If (L > 1, the series diverges.
 If (L = 1,) the test fails.
 Let (L = \lim_{n \to \infty} n\left\frac{a_{n+1}}{a_n}  1\right).

Comparison test (NCERT Class 12, Chapter 9, Section 9.10):
 If the series (a_n) and (b_n) are positive and (a_n \le b_n) for all (n,) then
 If (b_n) converges, (a_n) also converges.
 If (a_n) diverges, (b_n) also diverges.
 NCERT Examples: Solved Examples 9.10 (a), (b), (c), (d), (e), (f)
 If the series (a_n) and (b_n) are positive and (a_n \le b_n) for all (n,) then

Limit comparison test (NCERT Class 12, Chapter 9, Section 9.11):
 Let (L = \lim_{n \to \infty} \frac{a_n}{b_n}). If (L) exists and is finite and nonzero, then the series (a_n) and (b_n) either both converge or both diverge.

Integral test (NCERT Class 12, Chapter 9, Section 9.12):
 If (f(x)) is a positive, continuous, and decreasing function of (x) for (x \ge a), then
 If the improper integral (\int_a^\infinfiy f(x) dx) converges, then the series (a_n = f(n)) converges.
 If the improper integral (\int_a^\infinfiy f(x) dx) diverges, then the series (a_n = f(n)) diverges.
 NCERT Examples: Solved Examples 9.12 (a), (b), (c)
 If (f(x)) is a positive, continuous, and decreasing function of (x) for (x \ge a), then
4. Alternating Series

Leibniz’s alternating series test (NCERT Class 12, Chapter 9, Section 9.13):
 Let (a_n) be a positive, decreasing sequence of real numbers. Then the series (a_1  a_2 + a_3  \cdots + (1)^{n1} a_n), or ( \sum_{n=1}^\infty (1)^{n1} a_n), converges.
 NCERT Examples: Solved Examples 9.13 (a), (b), (c), (d), (e)

Estimation of the remainder for alternating series:
 The error or remainder ( R_n ) of an alternating series ( \sum_{n=1}^\infty (1)^{n1} a_n ) is estimated as $$R_n \le a_{n+1}.$$
5. Power Series
 Definition and convergence of power series (NCERT Class 11, Chapter 9, Section 9.4):
 A power series is an infinite series of the form (\sum_{n=0}^\infinf