monotone

(adjective)

property of a function to be either always decreasing or always increasing

Related Terms

  • Cauchy sequence

Examples of monotone in the following topics:

  • Alternating Series

    • The theorem known as the "Leibniz Test," or the alternating series test, tells us that an alternating series will converge if the terms ana_na​n​​ converge to 000 monotonically.
    • Proof: Suppose the sequence ana_na​n​​ converges to 000 and is monotone decreasing.
    • Since ana_na​n​​ is monotonically decreasing, the terms are negative.
    • an=1na_n = \frac1na​n​​=​n​​1​​ converges to 0 monotonically.
  • The Mean Value Theorem, Rolle's Theorem, and Monotonicity

  • The Integral Test and Estimates of Sums

    • Consider an integer NNN and a non-negative function fff defined on the unbounded interval [N,∞)[N, \infty )[N,∞), on which it is monotonically decreasing.
    • The above examples involving the harmonic series raise the question of whether there are monotone sequences such that f(n)f(n)f(n) decreases to 000 faster than 1n\frac{1}{n}​n​​1​​but slower than 1n1+ε\frac{1}{n^{1 + \varepsilon}}​n​1+ε​​​​1​​ in the sense that:
  • Tips for Testing Series

    • Integral test: For a positive, monotone decreasing function f(x)f(x)f(x) such that f(n)=anf(n)=a_nf(n)=a​n​​, if ∫1∞f(x)dx=limt→∞∫1tf(x)dx<∞\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty∫​1​∞​​f(x)dx=lim​t→∞​​∫​1​t​​f(x)dx<∞ then the series converges.
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