what does the sigma notation look like for the sequence 1/4,-1/9,1/16,-1/25
The denominator is i², so the term is 1/i².
The signs are alternating, so multiply each term by (-1)^i to get
5
∑ (-1)^i / i²
i=2
Nowadays, students are very likely to avoid STEM, especially math. Students should take math from school level so that they don't have to face any problem in future. Schools also should take the matter seriously. Unfortunately most of the public schools do not do so.
The sigma notation for the given sequence can be written as:
∑ (n = 1 to ∞) (-1)^(n+1) / n^2
Here, ∑ denotes the sum, n represents the index of the term, and the expression (-1)^(n+1) / n^2 is the general term of the sequence. The index n starts at 1 and goes to infinity, indicating that we are summing up all the terms in the given sequence.
The sigma notation is a way to represent the sum of a sequence of terms using the Greek letter sigma (∑). To find the sigma notation for the given sequence 1/4, -1/9, 1/16, -1/25, we need to consider the pattern of the terms.
The given sequence alternates between positive and negative fractions, with the denominator being a perfect square. We can observe that each term has a positive or negative sign, which is determined by the position of the term in the sequence.
Let's break down the sigma notation step by step:
1. Start with the sigma symbol (∑).
2. Below the sigma symbol, write the starting value of the index of summation. In this case, let's use "n = 1" since the sequence starts at the first term.
3. Above the sigma symbol, write the final value of the index of summation. Since there are four terms in the sequence, we can use "n = 4" as the final value.
4. Inside the sigma symbol, write the expression for the terms of the sequence.
- The numerator of the terms is always 1.
- The denominator is a perfect square, and it corresponds to the index n. So, the denominator can be written as n^2.
- Since each term alternates between positive and negative, we can include (-1)^(n+1) to represent this change in sign.
Putting it all together, the sigma notation for the given sequence is:
∑ ((-1)^(n+1) / n^2) from n = 1 to n = 4.
This notation represents the sum of the sequence 1/4, -1/9, 1/16, -1/25, where each term is alternately positive and negative, with the denominator being a perfect square.