# Basic Concepts of sequences and series

### Sequences:

Definition:

• A sequence is an ordered list of numbers arranged according to a certain rule or pattern.

Types of Sequences:

1. Arithmetic Sequence:

• Defined by a common difference (d) between consecutive terms.
• General form: ${a}_{n}={a}_{1}+\left(n-1\right)\cdot d$ where ${a}_{n}$ is the $n$th term.
2. Geometric Sequence:

• Defined by a common ratio (r) between consecutive terms.
• General form: ${a}_{n}={a}_{1}\cdot {r}^{\left(n-1\right)}$ where ${a}_{n}$ is the $n$th term.
3. Harmonic Sequence:

• Defined by the reciprocals of an arithmetic sequence.
• General form: ${a}_{n}=\frac{1}{{a}_{1}+\left(n-1\right)\cdot d}$ where ${a}_{n}$ is the $n$th term.

Properties of Sequences:

• Convergent vs. Divergent:

• Convergent sequences approach a specific limit as $n$ approaches infinity.
• Divergent sequences do not approach a limit and may tend towards positive or negative infinity.
• Monotonic Sequences:

• Monotonic sequences either consistently increase (monotonically increasing) or decrease (monotonically decreasing) with each term.

### Series:

Definition:

• A series is the sum of the terms in a sequence.

Types of Series:

1. Finite Series:

• The sum of a specific number of terms in a sequence.
• Example: ${S}_{n}={a}_{1}+{a}_{2}+{a}_{3}+\mathrm{.}\mathrm{.}\mathrm{.}+{a}_{n}$
2. Infinite Series:

• The sum of an infinite number of terms in a sequence.
• Example: $S={a}_{1}+{a}_{2}+{a}_{3}+\mathrm{.}\mathrm{.}\mathrm{.}+{a}_{n}+\mathrm{.}\mathrm{.}\mathrm{.}$

Properties of Series:

• Convergence of Infinite Series:

• Convergent series have a finite sum.
• Divergent series have an infinite sum or no sum at all.
• Tests for Convergence:

• Divergence Test: If the terms of the series do not approach zero, the series diverges.
• Geometric Series Test: Determines convergence for geometric series based on the common ratio.
• Integral Test: Applies to series with non-negative, decreasing terms and connects the convergence of series to definite integrals.
• Comparison Test: Compares the given series with a known series to establish convergence or divergence.
• Summation of Series:

• Formulas exist for finding the sum of certain types of series, like arithmetic, geometric, and telescoping series.

### Common Formulas:

• Arithmetic Series Sum: ${S}_{n}=\frac{n}{2}\left({a}_{1}+{a}_{n}\right)$
• Geometric Series Sum: $S=\frac{{a}_{1}}{1-r}$ (for $\mathrm{\mid }r\mathrm{\mid }<1$)

### Applications:

• Mathematics: Used in calculus, discrete mathematics, and number theory.
• Physics: Applied in modeling various physical phenomena involving continuous change or discrete values.
• Computer Science: Used in algorithms, data structures, and cryptography.