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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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Geometric Series • Convergence
The Divergence Test • If the sequence {an} does not converge to 0, then the series diverges.
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
The p-test for Series • The p-series converges if and only if p > 1.