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 $a_n$ converge to $0$ monotonically.
    • Proof: Suppose the sequence $a_n$ converges to $0$ and is monotone decreasing.
    • Since $a_n$ is monotonically decreasing, the terms are negative.
    • $a_n = \frac1n$ converges to 0 monotonically.
  • The Mean Value Theorem, Rolle's Theorem, and Monotonicity

  • The Integral Test and Estimates of Sums

    • Consider an integer $N$ and a non-negative function $f$ defined on the unbounded interval $[N, \infty )$, on which it is monotonically decreasing.
    • The above examples involving the harmonic series raise the question of whether there are monotone sequences such that $f(n)$ decreases to $0$ faster than $\frac{1}{n}$but slower than $\frac{1}{n^{1 + \varepsilon}}$ in the sense that:
  • Tips for Testing Series

    • Integral test: For a positive, monotone decreasing function $f(x)$ such that $f(n)=a_n$, if $\int_{1}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{1}^{t} f(x)\, dx < \infty$ then the series converges.
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