monotone

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.

• The Role of Color

  • Some presenters change up their color schemes regularly to prevent their presentations from becoming too monotonous.

• Consumer Purchasing Behavior

  • Examples of Social Functions: Decisiveness, neutrality, anonymity, monotonicity, unanimity, homogeneity and weak and strong Pareto optimality.
  • Some specifications of the social functions are decisiveness, neutrality, anonymity, monotonicity, unanimity, homogeneity and weak and strong Pareto optimality.

• General Case

  • Thereafter, there is a monotonic decrease in the observed frequency as it gets closer to the observer, through equality when it is closest to the observer, and a continued monotonic decrease as it recedes from the observer.

• Comparing Two Populations: Independent Samples

  • It assesses how well the relationship between two variables can be described using a monotonic function.
  • If there are no repeated data values, a perfect Spearman correlation of $1$ or $-1$ occurs when each of the variables is a perfect monotone function of the other.

• Postmodern

  • Monotonous apartment blocks were seen as drab and undesirable.

• Free Version Control Systems

  • GNU Arch or Monotone (or BitKeeper in the proprietary world).
  • "monotone is a free distributed version control system. it provides a simple, single-file transactional version store, with fully disconnected operation and an efficient peer-to-peer synchronization protocol. it understands history-sensitive merging, lightweight branches, integrated code review and 3rd party testing. it uses cryptographic version naming and client-side RSA certificates. it has good internationalization support, has no external dependencies, runs on linux, solaris, OSX, and windows, and is licensed under the GNU GPL."