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How to Write a Proof in Calculus

Calculus is the first mathematics course where students are introduced to the idea that their answers to a problem are not self-evident and therefore, in some cases, require proofs. This concept is first introduced when evaluating infinite series for convergence or divergence. Due to their nature, the total value of infinite series are often not readily obtainable and therefore their convergence or divergence is determined in comparison to other series. This comparison utilizes mathematical induction, not to be confused with logical induction, to determine the properties of the series being considered.

Instructions

    • 1

      Examine the problem and classify it based on its form. For example, given the problem of finding the convergence or divergence of the following series, Σ (1 / n^(1/2)), it is quickly apparent that this is in the form of a P - series.

    • 2

      Compare the given series to a known series. Continuing the example, the given series is larger, at all points along its domain, than a known series, specifically the harmonic series Σ 1 / n.

    • 3

      State the relationship between the series and how that affects the convergence or divergence of the original series. The important aspect of this point is not the convergence or divergence --- this is simply one example --- but rather how the properties of the given problem must behave in comparison to the properties of a known problem, the known problem in this example being the divergence of the harmonic series. This relationship and properties must be explicitly stated. Given the example, the relationship and proof between these two series could be stated as such: "The given series is a P - series that is larger than the harmonic series along its entire domain and the harmonic series is a divergent series; therefore, the given series must also diverge."

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