This hypothesis is right about operators on convergent and divergent series?

In summary, the conversation discusses the incorrect assumption of convergence in a series and the importance of understanding convergent and divergent series and their operations. This error is evident in the second image, where the series is treated as real numbers instead of infinity.
  • #1
MAGNIBORO
106
26
Sorry for the bad English , do not speak the language very well.
I posted this to know if the statement or " hypothesis " is correct .

thank you very much =D.

First Image:https://gyazo.com/7248311481c1273491db7d3608a5c48e
Second Image:https://gyazo.com/d8fc52d0c99e0094a6a6fa7d0e5273b6
Third Image:https://gyazo.com/035813f059d3eb7bb4131af7cd8f29c3
 
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  • #2
No, it isn't correct. Learn about convergent and divergent series, and which operations are allowed on them. You need to be very careful with convergence issues.
 
  • #3
micromass said:
No, it isn't correct. Learn about convergent and divergent series, and which operations are allowed on them. You need to be very careful with convergence issues.
the error is on the part of the two match series with a formula? ( image 2 )
 
  • #4
MAGNIBORO said:
the error is on the part of the two match series with a formula? ( image 2 )

The error is assuming the series converges and thus can be treated as real numbers instead of infinity.
 
  • #5
micromass said:
The error is assuming the series converges and thus can be treated as real numbers instead of infinity.
thanks
 

1. What is the difference between convergent and divergent series?

A convergent series is a series in which the sum of its terms approaches a single finite value as the number of terms increases. In contrast, a divergent series is a series in which the sum of its terms either approaches infinity or does not have a limit.

2. How do operators affect convergent and divergent series?

Operators, such as addition, subtraction, multiplication, and division, can be applied to both convergent and divergent series. However, the result may differ depending on the specific series and operator used. In some cases, applying an operator to a convergent series may result in a divergent series.

3. Can a convergent series become a divergent series after applying an operator?

Yes, it is possible for a convergent series to become a divergent series after applying an operator. For example, if an operator is applied to a convergent series in a way that the resulting series has terms that increase without bound, then the series becomes divergent.

4. Is it possible for a divergent series to become a convergent series after applying an operator?

No, a divergent series cannot become a convergent series after applying an operator. The definition of a divergent series is that its terms either approach infinity or do not have a limit, so no matter what operator is applied, the series will still exhibit this behavior.

5. How can I determine if an operator will result in a convergent or divergent series?

The result of an operator on a series can be determined by using mathematical techniques such as the ratio test or the comparison test. These tests can help determine if a series is convergent or divergent, and how an operator may affect the series. Additionally, understanding the properties of the specific operator being used can also provide insight into the behavior of the resulting series.

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