Question 1

What is sequence convergence?

Accepted Answer

Sequence convergence occurs when the terms of a sequence approach a specific limit as the index approaches infinity. A sequence {aₙ} converges to L if lim(n→∞) aₙ = L, meaning the terms get arbitrarily close to L for sufficiently large n.

Question 2

How do you test for sequence convergence?

Accepted Answer

Common convergence tests include: 1) Limit test - check if lim(n→∞) aₙ exists and is finite. 2) Divergence test - if lim(n→∞) aₙ ≠ 0, the series diverges. 3) Comparison test - compare with known convergent/divergent sequences. 4) Ratio test - check lim(n→∞) |aₙ₊₁/aₙ|. 5) Root test - check lim(n→∞) |aₙ|^(1/n).

Question 3

What's the difference between sequence and series convergence?

Accepted Answer

Sequence convergence refers to individual terms approaching a limit: lim(n→∞) aₙ = L. Series convergence refers to the sum of terms approaching a limit: lim(n→∞) Σaₙ = S. A sequence can converge while its series diverges (like 1/n), or both can converge (like 1/n²).

Question 4

What are common convergent sequences?

Accepted Answer

Common convergent sequences include: 1/n² → 0 (p-series with p>1), n/(n+1) → 1 (rational), (-1)ⁿ/n → 0 (alternating), sin(n)/n → 0 (bounded), ln(n)/n → 0 (logarithmic), eⁿ/n! → 0 (factorial dominates exponential). Each has specific convergence properties and tests.

Question 5

How do you find the limit of a sequence?

Accepted Answer

To find sequence limits: 1) Identify the sequence type (rational, exponential, trigonometric, etc.). 2) Apply appropriate techniques: L'Hôpital's rule for ∞/∞ forms, squeeze theorem for bounded sequences, algebraic manipulation for rational sequences. 3) Check if the limit exists and is finite. 4) Verify convergence using appropriate tests.

Question 6

What is the divergence test?

Accepted Answer

The divergence test states that if lim(n→∞) aₙ ≠ 0, then the series Σaₙ diverges. However, if lim(n→∞) aₙ = 0, the test is inconclusive - the series may converge or diverge. This is why we need other tests like comparison, ratio, or integral tests.

Question 7

How do you use the comparison test?

Accepted Answer

The comparison test compares your sequence with known sequences: If 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges. If 0 ≤ bₙ ≤ aₙ and Σbₙ diverges, then Σaₙ diverges. Common comparison sequences include p-series (1/nᵖ) and geometric series (rⁿ).

Question 8

What is the ratio test for sequences?

Accepted Answer

The ratio test examines lim(n→∞) |aₙ₊₁/aₙ|: If the limit < 1, the series converges absolutely. If the limit > 1, the series diverges. If the limit = 1, the test is inconclusive. This test is particularly useful for sequences involving factorials, exponentials, or powers.

Question 9

How do you analyze alternating sequences?

Accepted Answer

For alternating sequences like (-1)ⁿaₙ, use the alternating series test: If |aₙ| is decreasing and lim(n→∞) |aₙ| = 0, then the series converges. The alternating harmonic series (-1)ⁿ/n converges, while the regular harmonic series 1/n diverges.

Question 10

What are practical applications of sequence convergence?

Accepted Answer

Sequence convergence is crucial in: numerical analysis (iterative methods), physics (wave functions, quantum mechanics), engineering (signal processing, control systems), computer science (algorithms, machine learning), economics (compound interest, population models), and calculus (Taylor series, Fourier series).

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