Help with alternating series sum

In summary, the conversation is discussing finding the sum of the series 1 - e + e2/2! - e3/3! + e4/4! + ... using the MacLaurin series for ex and determining that the value of x needed to get the alternate series is -e.
  • #1
Abyssnight
5
0

Homework Statement



Given the following: 1 - e + e2/2! - e3/3! + e4/4! + ...
Find the sum of series

Homework Equations



The MacLaurin equation for ex

The Attempt at a Solution



Well I thought that it would look like [tex]\sum(-1)^n\frac{e^n}{n!}[/tex]
Tried the Ration Test and got no where. So I'm just kind of stumped
 
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  • #2
Consider the MacLaurin series for ex.

ex = 1 + x + x2/2! + x3/3! + x4/4! ...

Now, what do you think you need to put into x to get the alternate series in your question?
 
  • #3
Wow, haha. I must have had a long night to for some reason miss the obvious. x = -e and it works. Thank you haha.
 

1. What is an alternating series sum?

An alternating series sum is a type of mathematical series where the terms alternate in sign (positive and negative). This means that the terms in the series are either added or subtracted in an alternating pattern. For example, an alternating series sum may look like 1 - 2 + 3 - 4 + 5 - 6 + ...

2. How do you determine the convergence of an alternating series sum?

To determine the convergence of an alternating series sum, you can use the Alternating Series Test. This test states that if the terms in an alternating series decrease in magnitude and approach 0 as n increases, then the series will converge. This means that the series will have a finite sum and will not continue to increase indefinitely.

3. Can an alternating series sum diverge?

Yes, an alternating series sum can diverge. If the terms in the series do not decrease in magnitude and approach 0 as n increases, then the series will not converge. This means that the series will either have an infinite sum or will oscillate between positive and negative values without approaching a specific value.

4. How do you find the sum of an alternating series?

To find the sum of an alternating series, you can use the formula for the sum of an infinite geometric series. This formula is a/(1-r), where a is the first term and r is the common ratio. In an alternating series, the common ratio is -1. However, if the series does not meet the necessary conditions for convergence, the sum cannot be determined.

5. Are there any other tests that can be used to determine the convergence of an alternating series sum?

Yes, there are other tests that can be used, such as the Ratio Test and the Root Test. These tests can be used to determine the convergence of a wider range of series, including alternating series. However, the Alternating Series Test is specifically designed for alternating series and may be more efficient in determining convergence for this type of series.

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