A series, when viewed as a sequence of partial sums, is defined by the sequence formed by successively adding terms of a given sequence.
Let's break this down further:
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Sequence: Start with an infinite sequence of numbers:
a₁, a₂, a₃, ..., aₙ, ... -
Partial Sums: Create a new sequence by summing the first term, the first two terms, the first three terms, and so on. These are the partial sums:
S₁ = a₁S₂ = a₁ + a₂S₃ = a₁ + a₂ + a₃Sₙ = a₁ + a₂ + a₃ + ... + aₙ- ... and so on.
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The Series as a Sequence: The series, denoted by Σaₙ (sum of all aₙ), is then represented by the sequence of these partial sums:
S₁, S₂, S₃, ..., Sₙ, ...This sequence of partial sums is crucial for determining whether the series converges or diverges. -
Convergence and Divergence:
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Convergence: If the sequence of partial sums
S₁, S₂, S₃, ..., Sₙ, ...approaches a finite limit L as n approaches infinity, then the series is said to converge to L, and we write Σaₙ = L. In other words, lim (n→∞) Sₙ = L. -
Divergence: If the sequence of partial sums does not approach a finite limit (it either goes to infinity, negative infinity, or oscillates), then the series is said to diverge.
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Example:
Consider the sequence aₙ = 1/2ⁿ.
- a₁ = 1/2
- a₂ = 1/4
- a₃ = 1/8
- ...
The series is Σ(1/2ⁿ). Let's find the partial sums:
- S₁ = 1/2
- S₂ = 1/2 + 1/4 = 3/4
- S₃ = 1/2 + 1/4 + 1/8 = 7/8
- S₄ = 1/2 + 1/4 + 1/8 + 1/16 = 15/16
- ...
- Sₙ = 1 - (1/2)ⁿ
As n approaches infinity, Sₙ approaches 1. Therefore, the series Σ(1/2ⁿ) converges to 1. The sequence of partial sums (1/2, 3/4, 7/8, 15/16, ...) gets closer and closer to 1.
In summary, defining a series as a sequence of its partial sums provides a rigorous way to analyze its convergence or divergence. Instead of directly trying to "sum" an infinite number of terms (which can be tricky), we examine the behavior of the sequence formed by the successive additions of those terms.
