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Sequences:
- A sequence is an ordered list of numbers called terms. The terms follow a specific pattern or rule.
- Sequences can be finite (having a limited number of terms) or infinite (continuing indefinitely).
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Arithmetic Sequence:
- In an arithmetic sequence, each term is found by adding or subtracting a common difference (d) to the previous term.
- The nth term of an arithmetic sequence can be represented as: aₙ = a₁ + (n-1)d, where a₁ is the first term.
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Geometric Sequence:
- In a geometric sequence, each term is found by multiplying or dividing the previous term by a common ratio (r).
- The nth term of a geometric sequence can be represented as: aₙ = a₁ × r^(n-1), where a₁ is the first term.
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Series:
- A series is the sum of the terms of a sequence. It can be finite or infinite.
- The sum of the first n terms of a sequence is called an n-th partial sum.
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Arithmetic Series:
- An arithmetic series is the sum of the terms of an arithmetic sequence.
- The sum of the first n terms of an arithmetic series (Sₙ) can be calculated using the formula: Sₙ = n/2(a₁ + aₙ).
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Geometric Series:
- A geometric series is the sum of the terms of a geometric sequence.
- The sum of the first n terms of a geometric series (Sₙ) can be calculated using the formula: Sₙ = a₁(1 - rⁿ)/(1 - r), where r is the common ratio.
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Convergence and Divergence:
- A series converges if the sum of its terms approaches a finite value as the number of terms increases.
- A series diverges if the sum of its terms does not approach a finite value as the number of terms increases.
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