Sequences and Series

Sequences and Series

1. Definitions and Key Concepts

• Sequence: An ordered list of numbers following a specific pattern (e.g., $a_1, a_2, a_3, \dots$).
• Series: The sum of terms in a sequence (e.g., $S_n = a_1 + a_2 + \dots + a_n$).

2. Types of Sequences

1. Arithmetic Sequence

   • Each term differs by a constant $d$ (common difference).
   • General term: $a_n = a_1 + (n-1)d$.
   • Example: $2, 5, 8, 11, \dots$ ($d = 3$).

2. Geometric Sequence

   • Each term is multiplied by a constant $r$ (common ratio).
   • General term: $a_n = a_1 \cdot r^{n-1}$.
   • Example: $3, 6, 12, 24, \dots$ ($r = 2$).

3. Special Sequences

   • Fibonacci: $F_n = F_{n-1} + F_{n-2}$, with $F_1 = 1, F_2 = 1$.
   • Quadratic/Cubic: Defined by polynomial rules (e.g., $a_n = n^2$).

3. Types of Series

1. Finite Series

   • Sum of a limited number of terms.
   • Arithmetic Series Sum: $S_n = \frac{n}{2}(2a_1 + (n-1)d)$.
   • Geometric Series Sum: $S_n = a_1 \frac{1-r^n}{1-r}$ ($r \neq 1$).

2. Infinite Series

   • Sum of infinitely many terms (converges if sum approaches a finite limit).
   • Geometric Series Convergence: $S = \frac{a_1}{1-r}$ (if $|r| < 1$).

4. Convergence Tests for Infinite Series

• Divergence Test: If $\lim_{n \to \infty} a_n \neq 0$, the series diverges.
• Integral Test: If $f(n) = a_n$ is positive and decreasing, $\sum a_n$ converges iff $\int f(x)dx$ converges.
• Comparison Test: Compare to a known convergent/divergent series.
• Ratio Test: For $\sum a_n$, if $\lim |a_{n+1}/a_n| < 1$, it converges.

5. Power Series and Taylor Expansions

• Power Series: $\sum_{n=0}^\infty c_n (x-a)^n$.
• Taylor Series: Represents functions as infinite sums:
  $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n.$
  • Maclaurin Series (case where $a = 0$):
    Example: $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$.

6. Applications

• Finance: Compound interest (geometric series).
• Physics: Fourier series for wave analysis.
• Computer Science: Algorithm analysis (summing operation counts).

7. Key Formulas

• Sum of First $n$ Natural Numbers: $\sum_{k=1}^n k = \frac{n(n+1)}{2}$.
• Sum of Squares: $\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$.
• Harmonic Series: $\sum_{k=1}^\infty \frac{1}{k}$ diverges.

8. Practical Problem-Solving Steps

1. Identify the pattern (arithmetic/geometric/other).
2. Write the general term $a_n$.
3. For series, apply summation formulas or tests for convergence.

Sequences and series form the backbone of calculus, discrete mathematics, and analytical problem-solving, with wide-ranging theoretical and practical implications.