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bincy
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View attachment 145 where M>=2. A close upper bound also will be useful(not like 1 as the upper bound). Thanks in advance.This is also QPochhammer[1/M,1/M,inf]. Courtesy to mathematica.
bincybn said:View attachment 145 where M>=2. A close upper bound also will be useful(not like 1 as the upper bound). Thanks in advance.This is also QPochhammer[1/M,1/M,inf]. Courtesy to mathematica.
bincybn said:View attachment 145 where M>=2. A close upper bound also will be useful(not like 1 as the upper bound). Thanks in advance.This is also QPochhammer[1/M,1/M,inf]. Courtesy to mathematica.
You can get it bounded away from 0 (for any $m>1$) like this. First, for $0<x<1$, $$ -\ln(1-x) = x + \tfrac{x^2}2 + \tfrac{x^3}3 + \ldots < x + x^2 + x^3 + \ldots = \tfrac x{1-x}.$$bincybn said:Can you tell anything about the lower bound? My doubt is whether it will converge to zero or not?
The limit of a product term is a mathematical concept that describes the behavior of the product of two or more functions as their input values approach a certain value or "limit". It is commonly denoted as lim f(x)g(x) as x approaches a certain value.
The limit of a product term can be calculated by evaluating the limit of each individual function and then multiplying the resulting limits together. In other words, lim f(x)g(x) = lim f(x) * lim g(x) as x approaches a certain value.
Understanding limits of product terms is essential in many areas of mathematics and science, particularly in calculus and related fields. It allows us to analyze the behavior of functions and make predictions about their values as their inputs approach certain values.
No, if the limit of at least one of the individual functions does not exist, then the limit of the product term also does not exist. However, it is possible for the individual limits to exist while the limit of the product term does not, if there is a discontinuity in the product term.
Yes, there are a few special cases that can arise when finding the limit of a product term. These include when one or more of the functions involved is a constant, when one or more of the functions approaches infinity, and when the functions involve trigonometric or exponential functions.