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.

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1. Classical Progressions Summary

📊Visual Mathematical Intuition

Infinite Geometric Series Dissection & Telescoping Cascade

Infinite Sum:$S_\infty = \sum_{k=1}^\infty \left(\frac{1}{2}\right)^k = \frac{1/2}{1 - 1/2} = 1$

Telescoping Ladder:$\sum_{k=1}^n (V_k - V_{k-1}) = V_n - V_0$

Visual unit-square dissection demonstrating $\sum_{k=1}^\infty (1/2)^k = 1$, alongside the telescoping cascade ladder collapsing $S_n = V_n - V_0$.

Arithmetic Progression (AP)

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

Geometric Progression (GP)

• General term: $a_n=a{r}^{n-1}$
• Sum of $n$ terms: $S_n=\frac{a(1-r^n)}{1-r}$ ($r\ne 1$)
• Sum of infinite GP ($|r|<1$): $S_{\mathrm{\infty }}=\frac{a}{1-r}$

Arithmetic-Geometric Progression (AGP)

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

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

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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:

$$S_n=\sum _{r=1}^n{T}_r=(V_1-V_0)+(V_2-V_1)+\cdots +(V_n-V_{n-1})=V_n-V_0$$

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 $(r+k)-r=k$:

$$T_r=\frac{1}{k}[\frac{1}{r(r+1)\dots (r+k-1)}-\frac{1}{(r+1)(r+2)\dots (r+k)}]=V_{r-1}-V_r$$$$\sum _{r=1}^{\mathrm{\infty }}{T}_r=\frac{1}{k\cdot 1\cdot 2\dots k}$$

Type II: Factors in Numerator

$$T_r=r(r+1)(r+2)\dots (r+k)=\frac{1}{k+2}[r(r+1)\dots (r+k)(r+k+1)-(r-1)r(r+1)\dots (r+k)]$$$$S_n=\frac{n(n+1)(n+2)\dots (n+k+1)}{k+2}$$

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3. Standard Sigma Sum Formulas

$$\sum _{k=1}^{n}k=\frac{n(n+1)}{2},\sum _{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6},\sum _{k=1}^n{k}^3={\left[\frac{n(n+1)}{2}\right]}^2$$

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4. Double Summations $\sum \sum$

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

$$\sum _{i=1}^n\sum _{j=1}^n{a}_i{a}_j=\left(\sum _{i=1}^n{a}_i\right)\left(\sum _{j=1}^n{a}_j\right)={\left(\sum _{i=1}^n{a}_i\right)}^2$$

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

$$\sum _{1\le i<j\le n}{a}_i{a}_j=\frac{1}{2}[{\left(\sum _{i=1}^n{a}_i\right)}^2-\sum _{i=1}^n{a}_i^2]$$