Write in sigma notation 1/3,2/9,-7/27,14/81,-23/243,34/729...
Tricky. Note that the 1st term should be minus, but it is not. But it can be written as (-1)/(-3)
With that in mind, the numerators form the sequence
where the 1st differences are
3,5,7,9,11 and the second differences are 2, meaning the numerators are a quadratic function: n^2-2
so the sum is
∑ (-1)^n * (n^2-2)/3^n
To write the given series in sigma notation, we need to find the pattern in terms of an expression involving the variable n. Let's analyze the given sequence:
1/3, 2/9, -7/27, 14/81, -23/243, 34/729...
By observing the numerator and denominator values, we can determine that the sign of each term alternates between positive and negative. Additionally, the numerators form a pattern of successive odd numbers, while the denominators increase as powers of 3, starting from 3^1.
Therefore, we can express the series in sigma notation as follows:
∑ ((-1)^(n-1) * (2n - 1))/(3^n)
Where n represents the position (starting from 1) in the sequence.
To write the given sequence of fractions in sigma notation, you need to find a pattern in the numerator and denominator. Let's observe the pattern:
1/3, 2/9, -7/27, 14/81, -23/243, 34/729, ...
Looking at the numerators, we can see that they are alternating between positive and negative values, starting with 1, then 2, then -7, then 14, and so on.
Now, let's observe the denominators. Each denominator is the power of 3, starting with 3^1, then 3^2, then 3^3, then 3^4, and so on.
We can use this pattern within a summation symbol to write the series in sigma notation. The summation symbol Σ represents a series or sum of terms. The index variable k represents the position of each term.
The numerator alternates between positive and negative values, so we will use (-1)^(k+1) as the sign of the numerator. The denominator is a power of 3, so we use 3^k.
Therefore, the given series can be written in sigma notation as:
∑ [(-1)^(k+1) * (k+1)/3^k]
This notation represents the sum of the terms where k ranges from 0 to infinity.