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Series Solutions of Differential Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
subject that the limits exist. If the series (6.1.2) converges only at the center for x=x0, we say that the series has zero radius of convergence or the series diverges. For example, the series ∑n⩾0n!xn diverges for any x other than x=0. Note that divergent series are important in many areas of applied mathematics, but they are not our concern in this chapter.
Infinite Series
Published in John Srdjan Petrovic, Advanced Calculus, 2020
This series is known as the Harmonic series, and it is a divergent series. Unfortunately, the Divergence Test is not of any help. Namely, an = 1/n → 0, and the test works only when the limit is not equal to 0. However, we have established in Example 2.6.5 that the sequence an=1+12+13+…+1n is not a Cauchy sequence. By Theorem 2.6.6, it is not convergent, so the Harmonic series diverges.
Transform methods
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Remark. The Bernoulli numbers also arise in connection with the Riemann zeta function. Assuming that Re(s) > 1, the formula ζ(s)=∑n=1∞1ns defines the zeta function. The method of analytic continuation leads to a function (still written ζ) that is complex analytic except at 1, where it has a pole. The formula ζ(−s)=−Bs+1s+1 holds when s is a positive integer. The strange value of −112 for Σn in Example 2.3 arises as −B22, by formally putting s = 1 in the series definition. This definition requires Re(s) > 1. The method of assigning values to divergent series in this manner occurs in both number theory and mathematical physics; the method is sometimes known as zeta function regularization.
Posing problems and solving self-generated problems: the case of convergence and divergence of series
Published in International Journal of Mathematical Education in Science and Technology, 2023
Özkan Ergene, Büşra Çaylan Ergene
Another difficulty based on series was the use of divergent and convergent series interchangeably. In the first of the two examples given below, the divergent series was used instead of the convergent series. In the second example, the convergent series was used instead of the divergent series. In a squash practice, the ball is thrown from 20 meters away. The ball travels twice as far each time it comes back. How far does the ball travel?A bee makes a certain amount of honey every day. It makes 1 gram on the first day, gram on the second day, gram on the third day … gram on the nth day. How many grams of honey does this bee make in total?
A note on a general divergent series
Published in International Journal of Mathematical Education in Science and Technology, 2023
In addition to the direct consequence of Theorem 2.1 regarding the divergence of the harmonic series, it can be shown that this theorem can be useful in checking the divergence of many slowly diverging series under the comparison test. For instance, let's create a divergent series that diverges more slowly than the harmonic series. For this, consider the following example.