Do the following diverge or converge? explain why

General
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do the following diverge or converge? explain why.

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Do the following diverge or converge? Explain why.

To determine whether a sequence or series diverges or converges, one must analyze its behavior as the number of terms approaches infinity. Here are some general guidelines and examples to help understand the concepts of convergence and divergence:

1. Sequences

A sequence is a list of numbers in a specific order. To determine if a sequence converges, we look at the limit of the sequence as n approaches infinity.

Example:
Consider the sequence a_n = \frac{1}{n}.

  • Convergence Analysis:
    • As n \to \infty, \frac{1}{n} \to 0.
    • Therefore, the sequence \frac{1}{n} converges to 0.

Example:
Consider the sequence b_n = n.

  • Divergence Analysis:
    • As n \to \infty, n \to \infty.
    • Therefore, the sequence n diverges because it grows without bound.

2. Series

A series is the sum of the terms of a sequence. To determine if a series converges, we examine the sum of its terms as the number of terms approaches infinity.

Example:
Consider the geometric series \sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n.

  • Convergence Analysis:
    • The series is geometric with a common ratio r = \frac{1}{2}.
    • A geometric series converges if |r| < 1.
    • Here, |\frac{1}{2}| < 1, so the series converges.
    • The sum of the series is \frac{1}{1 - r} = \frac{1}{1 - \frac{1}{2}} = 2.

Example:
Consider the harmonic series \sum_{n=1}^{\infty} \frac{1}{n}.

  • Divergence Analysis:
    • The harmonic series is known to diverge.
    • As n \to \infty, the partial sums grow without bound.
    • Therefore, the harmonic series diverges.

Tests for Convergence and Divergence

  1. Limit Test for Sequences:

    • If \lim_{n \to \infty} a_n = L (a finite number), the sequence converges to L.
    • If \lim_{n \to \infty} a_n = \infty or does not exist, the sequence diverges.

  2. nth-Term Test for Divergence (Series):

    • If \lim_{n \to \infty} a_n \neq 0, the series \sum a_n diverges.
    • If \lim_{n \to \infty} a_n = 0, the test is inconclusive (the series may converge or diverge).

  3. Comparison Test:

    • Compare the series to a known benchmark series.
    • If 0 \leq a_n \leq b_n and \sum b_n converges, then \sum a_n converges.
    • If a_n \geq b_n \geq 0 and \sum b_n diverges, then \sum a_n diverges.

  4. Ratio Test:

    • Consider \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L.
    • If L < 1, the series converges.
    • If L > 1 or L = \infty, the series diverges.
    • If L = 1, the test is inconclusive.

  5. Root Test:

    • Consider \lim_{n \to \infty} \sqrt[n]{|a_n|} = L.
    • If L < 1, the series converges.
    • If L > 1 or L = \infty, the series diverges.
    • If L = 1, the test is inconclusive.

Conclusion

To determine if a sequence or series diverges or converges, it is essential to analyze the limit of the sequence or the sum of the series using appropriate tests. Each test has specific conditions that must be met to draw a conclusion about convergence or divergence. By applying these methods, one can accurately determine the behavior of sequences and series.