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The series can be compared to an integral to establish convergence or divergence. Let f ( n ) = a n {\displaystyle f(n)=a_{n}} be a positive and monotonically decreasing function . If
When X n converges in r-th mean to X for r = 1, we say that X n converges in mean to X. When X n converges in r-th mean to X for r = 2, we say that X n converges in mean square (or in quadratic mean) to X. Convergence in the r-th mean, for r ≥ 1, implies convergence in probability (by Markov's inequality).
While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be a sequence of positive numbers.
The divergence of the harmonic series was proven by the medieval mathematician Nicole ... This is the default definition of convergence of a sequence. Nørlund means
This definition is technically called Q-convergence, short for quotient-convergence, and the rates and orders are called rates and orders of Q-convergence when that technical specificity is needed. § R-convergence , below, is an appropriate alternative when this limit does not exist.
As the D in MACD, "divergence" refers to the two underlying moving averages drifting apart, while "convergence" refers to the two underlying moving averages coming towards each other. Gerald Appel referred to a "divergence" as the situation where the MACD line does not conform to the price movement, e.g. a price low is not accompanied by a low ...
"Beta-convergence" on the other hand, occurs when poor economies grow faster than rich ones. Economists say that there is "conditional beta-convergence" when economies experience "beta-convergence" but conditional on other variables (namely the investment rate and the population growth rate) being held constant.
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.