Key Concepts of Sequences and Series

Sequences and series are essential concepts in Algebra and Trigonometry, focusing on patterns in numbers. They include arithmetic and geometric sequences, their sums, and the behavior of series, helping us understand how numbers relate and grow over time.

1. Arithmetic sequences

   • A sequence where each term is obtained by adding a constant difference to the previous term.
   • The general form is ( a_n = a_1 + (n-1)d ), where ( d ) is the common difference.
   • The first term is denoted as ( a_1 ) and ( n ) represents the term number.

2. Geometric sequences

   • A sequence where each term is obtained by multiplying the previous term by a constant ratio.
   • The general form is ( a_n = a_1 \cdot r^{(n-1)} ), where ( r ) is the common ratio.
   • The first term is denoted as ( a_1 ) and ( n ) represents the term number.

3. Arithmetic series

   • The sum of the terms of an arithmetic sequence.
   • The formula for the sum of the first ( n ) terms is ( S_n = \frac{n}{2} (a_1 + a_n) ) or ( S_n = \frac{n}{2} (2a_1 + (n-1)d) ).
   • It can be visualized as the area under a linear function.

4. Geometric series

   • The sum of the terms of a geometric sequence.
   • The formula for the sum of the first ( n ) terms is ( S_n = a_1 \frac{1 - r^n}{1 - r} ) (for ( r \neq 1 )).
   • If ( |r| < 1 ), the series converges to ( S = \frac{a_1}{1 - r} ) as ( n ) approaches infinity.

5. Convergent and divergent series

   • A convergent series has a finite sum as the number of terms approaches infinity.
   • A divergent series does not have a finite sum; it may approach infinity or oscillate.
   • Understanding convergence is crucial for evaluating infinite series.

6. Infinite geometric series

   • A specific type of series where the terms form a geometric sequence and continue indefinitely.
   • The sum converges to ( S = \frac{a_1}{1 - r} ) if ( |r| < 1 ).
   • If ( |r| \geq 1 ), the series diverges.

7. Sigma notation

   • A concise way to represent the sum of a sequence of terms.
   • The notation ( \Sigma_{i=1}^{n} a_i ) indicates the sum of terms ( a_1, a_2, \ldots, a_n ).
   • It simplifies the expression of series and is widely used in mathematics.

8. Recursive formulas

   • A formula that defines each term of a sequence based on one or more previous terms.
   • For example, an arithmetic sequence can be defined as ( a_n = a_{n-1} + d ).
   • Useful for generating terms without needing to know the explicit formula.

9. Explicit formulas

   • A formula that allows direct computation of any term in a sequence without referencing previous terms.
   • For arithmetic sequences, it is ( a_n = a_1 + (n-1)d ); for geometric sequences, ( a_n = a_1 \cdot r^{(n-1)} ).
   • Facilitates quick calculations of specific terms.

10. Partial sums

    • The sum of the first ( n ) terms of a sequence or series.
    • Denoted as ( S_n ) for a series, it helps in understanding the behavior of the series as ( n ) increases.
    • Important for analyzing convergence and divergence of series.