Can Integrating to Infinity Determine Convergence or Divergence of a Series?

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In summary, you can determine if a series converges or diverges by taking the integral from n=# to infinity. If the result is infinity, the series diverges. If it is a number, the series converges and the sum may or may not be the same as the number it converges to. However, this method only works if the terms of the series are decreasing and not all series can be integrated easily.
  • #1
somebodyelse5
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Am I allowed to simply take the integral from n=# to infinity to determine if the series converges or diverges?

If the answer is infinity then it diverges
If the answer is a number then it converges, is the sum this number or does it converge to this number? Or is the number it converges to the same as the sum?
 
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  • #2
the terms must be decreasing and it will only tell you whether or not it converges or diverges
 
  • #3
Yes, you can do this. If you integrate the sequence in the series to infinity, the integral and the series do the same thing, diverge or converge. However, the sums are different values if they converge. The only problem is that most series encountered can't be integrated easily or at all.
 

FAQ: Can Integrating to Infinity Determine Convergence or Divergence of a Series?

What is the purpose of a convergent/divergent test?

A convergent/divergent test is used to determine whether a given series is convergent (approaches a finite limit) or divergent (does not approach a finite limit). It is an important tool in calculus and real analysis for evaluating the behavior of infinite series.

How do you determine if a series is convergent or divergent?

To determine if a series is convergent or divergent, you can use various tests such as the ratio test, the root test, or the integral test. These tests compare the terms of a series to a known function or sequence to determine if the series behaves in a convergent or divergent manner.

What is the difference between a convergent and divergent series?

A convergent series is one that has a finite limit or sum, meaning that as you add more terms to the series, the total value approaches a specific number. A divergent series, on the other hand, does not have a finite limit or sum, and the value of the series continues to grow or oscillate as more terms are added.

Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. It can only be one or the other. If a series has a finite limit, it is convergent, and if it does not have a finite limit, it is divergent.

How are convergent/divergent tests used in real-world applications?

Convergent/divergent tests are used in a wide range of real-world applications, such as in physics and engineering, to analyze the behavior of infinite series and determine their convergence or divergence. They are also used in financial mathematics to study the growth and decay of investments over time.

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