Sequences, Series & Telescoping Sums

Overview

A sequence $\{a_n\}$ is a function from $\mathbb{N} \to \mathbb{R}$ (or $\mathbb{C}$). A series is the sequence of partial sums $S_n = \sum_{k=1}^n a_k$. In competitive mathematics, mastery lies in transforming unfamiliar summations into known algebraic identities, geometric series, or telescoping cancellations.

1. Classical Progressions Summary

Arithmetic Progression (AP)

General term: $a_{n} = a + (n - 1) d$
Sum of $n$ terms: $S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)$

Geometric Progression (GP)

General term: $a_{n} = a r^{n - 1}$
Sum of $n$ terms: $S_{n} = \frac{a (1 - r^{n})}{1 - r}$ ( $r \neq 1$ )
Sum of infinite GP ( $| r | < 1$ ): $S_{\infty} = \frac{a}{1 - r}$

Arithmetic-Geometric Progression (AGP)

General AGP: $S = a + (a + d) r + (a + 2 d) r^{2} + \dots + (a + (n - 1) d) r^{n - 1}$

Sum of infinite AGP ( $| r | < 1$ ): $S_{\infty} = \frac{a}{1 - r} + \frac{d r}{(1 - r)^{2}}$

2. The $V_{n} - V_{n - 1}$ Telescoping Method

Fundamental Telescoping Identity

If the general term $T_r$ of a series can be decomposed as $T_r = V_r - V_{r-1}$, then: $$S_n = \sum_{r=1}^n T_r = (V_1 - V_0) + (V_2 - V_1) + \dots + (V_n - V_{n-1}) = V_n - V_0$$

Type I: Factors in Denominator (Equally Spaced Factors)

T_{r} = \frac{1}{r (r + 1) (r + 2) \dots (r + k)}

Multiply and divide by the difference of outermost factors $(r + k) - r = k$ :

T_{r} = \frac{1}{k} [\frac{1}{r (r + 1) \dots (r + k - 1)} - \frac{1}{(r + 1) (r + 2) \dots (r + k)}] = V_{r - 1} - V_{r}

\sum_{r = 1}^{\infty} T_{r} = \frac{1}{k \cdot 1 \cdot 2 \dots k}

Type II: Factors in Numerator

T_{r} = r (r + 1) (r + 2) \dots (r + k) = \frac{1}{k + 2} [r (r + 1) \dots (r + k) (r + k + 1) - (r - 1) r (r + 1) \dots (r + k)]

S_{n} = \frac{n (n + 1) (n + 2) \dots (n + k + 1)}{k + 2}

3. Standard Sigma Sum Formulas

\sum_{k = 1}^{n} k = \frac{n (n + 1)}{2}, \sum_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}, \sum_{k = 1}^{n} k^{3} = {[\frac{n (n + 1)}{2}]}^{2}

4. Double Summations $\sum \sum$

For independent summations over a grid $1 \leq i \leq n, 1 \leq j \leq n$ :

\sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i} a_{j} = (\sum_{i = 1}^{n} a_{i}) (\sum_{j = 1}^{n} a_{j}) = {(\sum_{i = 1}^{n} a_{i})}^{2}

For summations over $1 \leq i < j \leq n$ :

\sum_{1 \leq i < j \leq n} a_{i} a_{j} = \frac{1}{2} [{(\sum_{i = 1}^{n} a_{i})}^{2} - \sum_{i = 1}^{n} a_{i}^{2}]

Sequences, Series & Telescoping Sums ​

1. Classical Progressions Summary ​

Arithmetic Progression (AP) ​

Geometric Progression (GP) ​

Arithmetic-Geometric Progression (AGP) ​

2. The Vn−Vn−1 Telescoping Method ​

Type I: Factors in Denominator (Equally Spaced Factors) ​

Type II: Factors in Numerator ​

3. Standard Sigma Sum Formulas ​

4. Double Summations ∑∑ ​