The Alternating Series Error Estimate

Therefore, the maximum amount of error for s4 compared to the infinite series is less than or equal to the absolute value of (S4-S3) which is equal to 1/17. In summary, the maximum amount of error for s4 compared to the infinite series is less than or equal to 1/17.
  • #1
xstetsonx
78
0
summation from 1 to infinite: (5n)/(n^2+1)
What is the maximum amount of error of s4 as compared to the infinite series?

error=S-Sn<bn+1



I got S4=-14/17
and S5=-25/26
but how you suppose to get S? i though you can't tell what a alternating series converge to
 
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  • #2
The key point about an "alternating series", with decreasing terms, is that the sum up to [itex]a_n[/itex] lies between the sum up to [itex]a_{n-1}[/itex] and the sum up to [itex]a_{n-2}[/itex]. And, therefore, the actual sum lies between any two consective partial sums. That is, the entire sum lies between [itex]S_4[/itex] and [itex]S_3[/itex] so the error cannot be larger than the distance between those two.
 

Related to The Alternating Series Error Estimate

What is the Alternating Series Error Estimate?

The Alternating Series Error Estimate is a mathematical tool used to approximate the value of a series that alternates between positive and negative terms. It helps to determine how close the approximation is to the actual value of the series.

How is the Alternating Series Error Estimate calculated?

The Alternating Series Error Estimate is calculated using the remainder term of a series, which is the difference between the actual value and the approximation. The formula for calculating the remainder term depends on the type of series and can be found in most calculus textbooks.

What is the significance of the Alternating Series Error Estimate?

The Alternating Series Error Estimate is important because it allows us to determine the accuracy of our approximations. By calculating the error estimate, we can determine if our approximation is close enough to the actual value or if we need to refine our estimation method.

Can the Alternating Series Error Estimate be used for all types of series?

No, the Alternating Series Error Estimate is specifically designed for alternating series, which are series that alternate between positive and negative terms. It cannot be used for other types of series, such as geometric or harmonic series.

How does the Alternating Series Error Estimate relate to the Alternating Series Test?

The Alternating Series Error Estimate is closely related to the Alternating Series Test, which is used to determine the convergence of an alternating series. The error estimate helps to determine the accuracy of the approximation used in the Alternating Series Test.

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