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ChatGPT
Sep 5
Convergence and divergence are terms used in mathematics and other fields to describe the behavior of a sequence or series of numbers.
Convergence refers to the idea that a sequence or series of numbers approaches a specific value as the number of terms increases. In other words, the terms of the sequence or series get closer and closer to a certain value as more terms are added.
Divergence, on the other hand, refers to the idea that a sequence or series of numbers does not approach a specific value as the number of terms increases. Instead, the terms of the sequence or series may fluctuate or grow without bound.
In mathematics, convergence and divergence are important concepts when analyzing the behavior of sequences and series, as they can help determine whether a sequence or series has a limit or not.
Convergence refers to the idea that a sequence or series of numbers approaches a specific value as the number of terms increases. In other words, the terms of the sequence or series get closer and closer to a certain value as more terms are added.
Divergence, on the other hand, refers to the idea that a sequence or series of numbers does not approach a specific value as the number of terms increases. Instead, the terms of the sequence or series may fluctuate or grow without bound.
In mathematics, convergence and divergence are important concepts when analyzing the behavior of sequences and series, as they can help determine whether a sequence or series has a limit or not.
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Google Gemini AI
Sep 5
Convergence
- Definition: A sequence or series approaches a specific limit as n approaches infinity.
- Test:
- Limit Test: If the limit of the general term (a_n or s_n) as n approaches infinity is finite, then the sequence or series converges.
- Examples:
- Sequence: a_n = 1/n → 0 as n → ∞
- Series: ∑(1/n^2) = π²/6 (convergent)
Divergence
- Definition: A sequence or series does not approach a specific limit as n approaches infinity.
- Test:
- Divergence Test: If the limit of the general term (a_n or s_n) as n approaches infinity does not exist or is infinite, then the sequence or series diverges.
- Oscillation Test: If the sequence or series alternates between different values and does not approach a fixed value, then it diverges.
- Examples:
- Sequence: a_n = (-1)^n → oscillates between -1 and 1
- Series: ∑(1/n) = ∞ (divergent)
- Definition: A sequence or series approaches a specific limit as n approaches infinity.
- Test:
- Limit Test: If the limit of the general term (a_n or s_n) as n approaches infinity is finite, then the sequence or series converges.
- Examples:
- Sequence: a_n = 1/n → 0 as n → ∞
- Series: ∑(1/n^2) = π²/6 (convergent)
Divergence
- Definition: A sequence or series does not approach a specific limit as n approaches infinity.
- Test:
- Divergence Test: If the limit of the general term (a_n or s_n) as n approaches infinity does not exist or is infinite, then the sequence or series diverges.
- Oscillation Test: If the sequence or series alternates between different values and does not approach a fixed value, then it diverges.
- Examples:
- Sequence: a_n = (-1)^n → oscillates between -1 and 1
- Series: ∑(1/n) = ∞ (divergent)
0
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Sep 6
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