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Understanding Convergence in Algebraic Geometric Series

Explore the convergence properties of geometric series in algebra, including the Divergence Test and the Harmonic Series Integral Test. Learn how the convergence of a sequence (an) relates to the divergence of the series. Discover the conditions under which the series and integral diverge or converge, and delve into the p-Test for Series, determining convergence based on the value of p. Enhance your understanding of algebra with detailed explanations and examples.

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Understanding Convergence in Algebraic Geometric Series

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  1. Geometric Series • Convergence

  2. Algebra with Series

  3. The Divergence Test • If the sequence {an} does not converge to 0, then the series diverges.

  4. The Harmonic Series

  5. Integral Test for a Positive Series • Suppose that, for all x 1, the function a(x) is continuous, positive and decreasing. Consider the series and the integralwhere ak = a(k) for integers k  1. • If either diverges, so does the other. • If either converges, so does the other. In this case, we have

  6. The p-test for Series • The p-series converges if and only if p > 1.

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