How to Write the Series 2, 4, 6, 8, ... to Sigma Notation

Writing the series of positive even numbers in sigma notation involves understanding both the pattern and the alternating sign. Let's break down the process step by step.

Understanding the Series

The given series is: 2, 4, 6, 8, ... This sequence consists of positive even numbers. The general term for this series can be represented as:

2i for i 1, 2, 3, ...

Sigma Notation without Alternation

Sigma notation, or the summation notation, is used to represent the sum of a series. For the given series, the sum of the first n positive even numbers can be written as:

[sum_{i1}^{n} 2i 2 4 6 ldots 2n]

Introducing Alternation: Positive and Negative Terms

Now, consider the need to alternate the signs of the terms, resulting in the pattern: 2 - 4 6 - 8 .... This can be achieved by multiplying the series with -1n-1. The general term for the series becomes 2(-1n-1)i -2(n - n/2) 2n, clearer is 2(-1n-1n).

[sum_{n1}^{infty} -1^{n-1} 2n 2 - 4 6 - 8 ldots]

Here, -1n-1 generates the alternating signs, with positive for odd n and negative for even n. Each term in the series is a positive even number, but the sign alternates as specified.

Alternative Views on Sigma Notation

Another way to express the alternating series is by starting from k 1 and using the term 2(-1k)k:

[sum_{k1}^{infty} 2(-1)^k k 2 - 4 6 - 8 ldots]

Each term here is alternately positive and negative, and the sequence starts from 2, as required.

Conclusion

In summary, to write the alternating series of positive even numbers in sigma notation, you must include the alternating sign pattern using the properties of -1n-1 or -1k. This allows for a clear representation of the series' behavior.

Key Takeaways: The general term for the series of positive even numbers is 2i. Each term in the alternating series can be represented as 2(-1n-1)i or 2(-1k)k. The alternating pattern is achieved using the -1n-1 or -1k term.