Series: estimate sum within .01

In summary, the question asks how many terms of the series 1/(1+n^2) must be added to estimate the sum within 0.01. By finding an upper bound for the error between the infinite sum and the partial sum of the first N terms, it can be determined that N must be at least 100 to make the error less than 0.01.
  • #1
rcmango
234
0

Homework Statement



How many terms of the series
infinity
E n =1

1/(1+n^2) must be added to estimate the sum within 0.01?

Homework Equations




The Attempt at a Solution



need help please. Also, the answer i believe it 100 terms. However i need to show work to support this answer.
 
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  • #2
If we denote the infinite sum by S, ie:

[tex]S=\sum_{n=1}^\infty \frac{1}{n^2+1} [/tex]

and the partial sum of the first N terms by SN:

[tex]S_N=\sum_{n=1}^N \frac{1}{n^2+1} [/tex]

Then the error induced by estimating the infinite sum by the partial sum of the first N terms is:

[tex]S-S_N=\sum_{n=N+1}^\infty \frac{1}{n^2+1} [/tex]

Can you find an upper bound for this sum? Here's a hint: 1/(n-1)-1/n=1/n(n-1).
 
  • #3
while plugging numbers into n in the equation, i can see that the equation appears to approach to 0.

whats next :)
 
  • #4
Yes, of course it approaches 0! "Next" is to answer your question: how large does n have to be to make it less than 0.01?
 

Related to Series: estimate sum within .01

1. What is the importance of estimating a sum within .01?

Estimating a sum within .01 is important because it allows us to get a close approximation of the actual sum without having to calculate the exact value. This can save time and effort, especially when dealing with large data sets.

2. How do you estimate a sum within .01?

To estimate a sum within .01, you can use a technique called rounding. This involves rounding each number in the sum to the nearest hundredth and then adding them together. For example, if the numbers are 3.456 and 2.789, you would round them to 3.46 and 2.79 and then add them to get an estimated sum of 6.25.

3. Can you use estimation to get an accurate sum within .01?

No, estimation will not give you an accurate sum within .01. It is only an approximation and may not be exact. However, it can give you a close enough estimate for practical purposes.

4. Are there any other methods for estimating a sum within .01?

Yes, there are other methods such as using averages or using a scientific calculator that allows you to specify the number of decimal places in your answer. However, rounding is the most commonly used method for estimating a sum within .01.

5. Why is it important to specify within .01 when estimating a sum?

Specifying within .01 is important because it sets a limit on the level of accuracy needed for the estimated sum. It allows us to focus on the numbers that are most significant and disregard any minor discrepancies. It also helps us to avoid errors in our estimation process.

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