Does the Maximum of Two Convergent Series Also Converge?

In summary, the Convergent Series Problem is a mathematical concept that involves determining if an infinite series will eventually reach a finite sum or continue to increase. The convergence of a series is determined by analyzing the behavior of its terms as the number of terms increases. Some common examples of convergent series include geometric series, telescoping series, and p-series. The convergence of a series is related to its sum, where a convergent series has a finite sum and a divergent series does not. The Convergent Series Problem is important in mathematics as it helps understand infinite series, has applications in calculus, and is crucial in other areas of mathematics such as complex analysis and number theory.
  • #1
porroadventum
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Homework Statement


Let Ʃ from n=1 to ∞ an and Ʃ from n=1 to ∞ bn be convergent series, with an[itex]\geq[/itex]0 and bn[itex]\geq[/itex]0 for all n[itex]\in[/itex][itex]N[/itex]. Show that the series Ʃ from n=1 to∞ max(an,bn) converges.



Homework Equations


I'm guessing it's got something to do with the cauchy criterrion for convergence of series but I'm not sure where to begin? Any hints would much appreciated
 
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  • #2
I would write

2 max(a,b)=a+b+|a-b|

or

max(a,b)<=a+b

Either of which obviously converge.
 

FAQ: Does the Maximum of Two Convergent Series Also Converge?

What is the Convergent Series Problem?

The Convergent Series Problem is a mathematical concept that involves determining whether an infinite series of numbers will eventually reach a finite sum, or if it will continue to increase without bound.

How is the convergence of a series determined?

The convergence of a series is determined by analyzing the behavior of its terms as the number of terms increases. If the terms approach a finite limit, the series is said to converge. If the terms do not approach a limit, the series is said to diverge.

What are some common examples of convergent series?

Some common examples of convergent series include the geometric series (1/2 + 1/4 + 1/8 + ...), the telescoping series (1 + 1/2 + 1/4 + 1/8 + ...), and the p-series (1 + 1/2^p + 1/3^p + ...), where p is a positive integer.

How is the convergence of a series related to its sum?

If a series is convergent, its sum is equal to the limit of the terms as the number of terms approaches infinity. However, if a series is divergent, it does not have a finite sum.

What is the importance of the Convergent Series Problem in mathematics?

The Convergent Series Problem is important in mathematics because it helps us understand the behavior of infinite series and their sums. It also has applications in calculus, where convergent series are used to approximate functions. Additionally, the concept of convergence is crucial in many other areas of mathematics, such as complex analysis and number theory.

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