Overview
This lecture covers the differences between sequences and series, key formulas for arithmetic and geometric sequences and series, recursive and explicit formulas, and how to solve related problems including using summation notation.
Sequences vs. Series
- A sequence is an ordered list of numbers separated by commas.
- A series is the sum of the terms in a sequence.
Arithmetic Sequences & Series
- Arithmetic sequence: each term increases by a constant difference (d), called the common difference.
- Formula for the nth term: ( a_n = a_1 + (n-1)d ).
- Recursive formula: ( a_1 ) is given, ( a_n = a_{n-1} + d ).
- Arithmetic series (sum of first n terms): ( S_n = \frac{n}{2}(a_1 + a_n) ).
Geometric Sequences & Series
- Geometric sequence: each term is multiplied by a constant ratio (r), called the common ratio.
- Formula for the nth term: ( a_n = a_1 \cdot r^{n-1} ).
- Recursive formula: ( a_1 ) is given, ( a_n = a_{n-1} \cdot r ).
- Geometric series sum (finite): ( S_n = a_1 \frac{1-r^n}{1-r} ).
- Geometric series sum (infinite): ( S = \frac{a_1}{1-r} ), only if ( -1 < r < 1 ).
Solving for Terms & Formulas
- To find unknowns in a sequence, set up equations using the known terms and solve for the unknowns.
- For arithmetic, solve for ( a_1 ) and ( d ); for geometric, solve for ( a_1 ) and ( r ).
Summation Notation (Sigma Notation)
- The Greek letter Sigma (( \Sigma )) indicates the sum over the specified range.
- Plug each value in the range into the formula and sum the results.
- Count terms by subtracting the lower index from the upper, then add one.
Convergent vs. Divergent Series
- A geometric series converges (sum exists) if ( -1 < r < 1 ); otherwise, it diverges.
Key Terms & Definitions
- Sequence — an ordered list of numbers.
- Series — the sum of terms of a sequence.
- Arithmetic Sequence — sequence where each term increases by a constant difference.
- Geometric Sequence — sequence where each term multiplies by a constant ratio.
- Common Difference (d) — the amount added in an arithmetic sequence.
- Common Ratio (r) — the amount multiplied in a geometric sequence.
- Recursive Formula — defines each term using previous term(s).
- Explicit Formula — directly computes the nth term.
- Sigma Notation (( \Sigma )) — notation for summing a series.
Action Items / Next Steps
- Practice using explicit and recursive formulas for both sequence types.
- Complete exercises using summation notation.
- Review homework on finding sums of finite and infinite geometric series.