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In mathematics, the nth-term test for divergence [1] is a simple test for the divergence of an infinite series: If ... Problems in Mathematical Analysis. American ...
This is also known as the nth-term test, test for divergence, or the divergence test. Ratio test This is ... is the Basel problem and the series converges to ...
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
It concerns sequences of integers in which each term is obtained from the previous term as follows: if a term is even, the next term is one half of it. If a term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence.
The n th partial sum S n is the sum of the first n terms of the sequence; that is, ... The reciprocals of square numbers produce a convergent series (the Basel problem):
This problem is commonly resolved by the use of spline interpolation. Here, the interpolant is not a polynomial but a spline : a chain of several polynomials of a lower degree. Interpolation of periodic functions by harmonic functions is accomplished by Fourier transform .
Problem-based learning (PBL) is a teaching method in which students learn about a subject through the experience of solving an open-ended problem found in trigger material. The PBL process does not focus on problem solving with a defined solution, but it allows for the development of other desirable skills and attributes.
Applications of the harmonic series and its partial sums include Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are needed to provide a complete range of responses, the connected components of random graphs, the block-stacking problem on how far over the edge ...