Convergent series. Is my logic correct?

In summary, a convergent series is a series where the sum of its terms approaches a finite limit as the number of terms increases. To determine if a series is convergent, various convergence tests can be used. A series can only be convergent or divergent, not both. Convergent series are important in mathematics for approximating functions, solving equations, and understanding sequences and series. Assuming a series is convergent based on its terms approaching zero is not always correct and proper convergence tests should be used.
  • #1
emilkh
7
0
Show [tex]
\sum_1^\infty\frac{x^n}{1+x^n} [/tex] converges when x is in [0,1)
[tex]
\sum_1^\infty\frac{x^n}{1+x^n} = \sum_1^\infty\frac{1}{1+x^n} * x^n <= \sum_1^\infty\frac{1}{1} * x^n = \sum_1^\infty x^n [/tex]

The last sum is g-series, converges since r = x < 1
 
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  • #2
Yes, that is correct. In fact, you can say more: the series converges for x in (-1, 1).
 
  • #3
ratio test?
 

FAQ: Convergent series. Is my logic correct?

What is a convergent series?

A convergent series is a series of numbers where the sum of their terms approaches a finite limit as the number of terms increases. In other words, the series reaches a specific value or "converges" instead of diverging to infinity.

How do you determine if a series is convergent?

To determine if a series is convergent, you can use various convergence tests such as the ratio test, the comparison test, or the integral test. These tests involve checking the behavior of the series' terms and comparing them to known convergent or divergent series.

Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent, not both. A series is considered convergent if its terms approach a finite limit, while a series is considered divergent if its terms do not approach a limit or approach infinity.

What is the importance of convergent series in mathematics?

Convergent series play a crucial role in mathematics, particularly in calculus and analysis. They are used to approximate functions, solve differential equations, and understand the behavior of sequences and series. They also have applications in physics, engineering, and other fields.

Is my logic correct if I assume a series is convergent if its terms approach zero?

No, assuming a series is convergent based on its terms approaching zero is not always correct. This is because there are cases where the terms of a series approach zero but the series still diverges, such as the harmonic series. It is important to use proper convergence tests to determine the convergence or divergence of a series.

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