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:

Arithmetic Sequence:
 Defined by a common difference (d) between consecutive terms.
 General form: ${a}_{n}={a}_{1}+(n1)\cdot d$ where ${a}_{n}$ is the $n$th term.

Geometric Sequence:
 Defined by a common ratio (r) between consecutive terms.
 General form: ${a}_{n}={a}_{1}\cdot {r}^{(n1)}$ where ${a}_{n}$ is the $n$th term.

Harmonic Sequence:
 Defined by the reciprocals of an arithmetic sequence.
 General form: ${a}_{n}=\frac{1}{{a}_{1}+(n1)\cdot d}$ 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:

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

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 nonnegative, 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: ${S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n})$
 Geometric Series Sum: $S=\frac{{a}_{1}}{1r}$ (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.