How Do I Calculate the Partial Sum of an Alternating Series?

In summary, An alternating series partial sum is the sum of a series in which the terms alternate between positive and negative numbers. To find the partial sum, determine the pattern of the series and add the first n terms. The alternating series test is used to determine convergence or divergence, and an alternating series can have an infinite number of terms and still converge. The error bound is the difference between the actual value and the partial sum and can be used to estimate accuracy.
  • #1
Nick_L
7
0
Can anyone help me out with calculating the partial sum of an alternating series? For example, how would I find the sum correct to 4 decimal places of:

CalcProblem.gif


What I tried was finding how many terms it would take the have an error that was < .0001 then found the sum with that many terms... I got 0.10969 as the partial sum using 4 terms.
 
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  • #2
Note that

[tex]\sum_{n=1}^{\infty}nx^n=\frac{x}{(x-1)^2}[/tex]

which can be obtained from the geometric series by computing the derivative and multiplying by x. Hence

[tex]\sum_{n=1}^{\infty}n(-\frac{1}{11})^n=-\frac{11}{144}[/tex]
 

FAQ: How Do I Calculate the Partial Sum of an Alternating Series?

What is an alternating series partial sum?

An alternating series partial sum is the sum of a series in which the terms alternate between positive and negative numbers. It is calculated by adding the first n terms of the series, where n is a positive integer.

How do you find the partial sum of an alternating series?

To find the partial sum of an alternating series, first determine the pattern of the series (e.g. -1, 2, -3, 4, -5, etc.) Then, add the first n terms of the series, where n is the number of terms you want to include in the partial sum.

What is the alternating series test?

The alternating series test is a test used to determine if an alternating series converges or diverges. It states that if the terms of an alternating series decrease in absolute value and approach 0, then the series converges. However, if the terms do not approach 0, the series diverges.

Can an alternating series have an infinite number of terms and still converge?

Yes, an alternating series can have an infinite number of terms and still converge. As long as the terms follow the criteria of the alternating series test (decreasing in absolute value and approaching 0), the series will converge.

What is the error bound for an alternating series?

The error bound for an alternating series is the difference between the actual value of the series and the partial sum. It is calculated by taking the absolute value of the next term in the series. The error bound can be used to estimate the accuracy of the partial sum.

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