Show that the series is convergent

In summary, the conversation is discussing how to show that the series Ʃ (-1)(n-1)/ √(n+3) converges and how to find the number of terms needed to add in order to have an error less than .001. The conversation mentions taking the derivative and setting up an inequality, but it is suggested to use the Alternating Series Test instead. The conversation ends with a discussion about how many terms are needed for the desired error, with differing opinions on the answer.
  • #1
knv
17
0
1. Show that the series is convergent and then find how many terms we need to add in order to find the sum with an error less than .001

Ʃ (-1)(n-1)/ √(n+3)

from n = 1---> infinity




2. I took the derivative.



3. f(x) = (x+3)-1/2
f'(x) = -1/2 (x+3)-3/2

Then I set up the following

Absolute value (1/(n+1+3)) < .001

n+4 > (1/.001)2

Got 999,997 for the answer. not sure what I am doing wrong. Help!

 
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  • #2


There really is no need to be taking derivatives here. You should try to apply the Alternating Series Test. If you don't know what that is, you can find it on Wikipedia. The page has all of your answers.
 
  • #3


would it be 999,998 terms ?
 
  • #4


knv said:
would it be 999,998 terms ?
How did you get that?

The absolute value of the 999,997th term is indeed 0.001 .

But the series is alternating and the absolute value of subsequent terms if decreasing. If we let Sk represent the kth partial sum, then I would expect the series to converge to a value very close to midway between Sk and Sk+1, for large k.
 

What does it mean for a series to be convergent?

When a series is convergent, it means that the sum of its terms approaches a finite value as the number of terms increases. In other words, as more terms are added to the series, the overall sum becomes closer and closer to a specific number.

How can you prove that a series is convergent?

To prove that a series is convergent, you can use various convergence tests such as the comparison test, ratio test, or the root test. These tests involve analyzing the behavior of the series' terms to determine if they approach zero or if they are bounded by another convergent series.

What happens if a series is not convergent?

If a series is not convergent, it means that the sum of its terms does not approach a finite value as the number of terms increases. This can happen if the terms in the series are increasing without bound, or if they are oscillating between positive and negative values.

Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent. If a series is convergent, it means that it approaches a finite value as the number of terms increases. If a series is divergent, it means that the sum of its terms does not approach a finite value.

What are some real-life applications of convergent series?

Convergent series are commonly used in mathematics, physics, and engineering to calculate values such as infinite sums, integrals, and probabilities. They are also used in financial and economic models to analyze trends and make predictions about future values.

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