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Summation notation is a concise way to represent the sum of a sequence of terms. Instead of writing out each term separately, we use the Greek letter Sigma (\text{\textbackslash sum}) to indicate the sum. This notation includes:

• The index of summation, typically denoted by a variable like n.
• The lower limit of summation, indicating where the summation starts.
• The upper limit of summation, indicating where the summation ends (or goes to infinity for an infinite series).

For instance, the series \(4 - 1 + \frac{1}{4} - \frac{1}{16} + \text{\textellipsis}\) can be written in summation notation as:
\text{\textbackslash sum}_{n=0}^{\text{\textbackslash infty}} 4 \text{\textlparen} \frac{-1}{4} \text{\textrparen}^n
This expression tells us to start from \(n=0\) and continue indefinitely, multiplying the first term, 4, by the common ratio raised to the power n at each step.

The common ratio in a geometric series is the constant factor by which each term is multiplied to get the next term. To find it, divide any term in the series by the previous term. In the given series \(4 - 1 + \frac{1}{4} - \frac{1}{16} + \text{\textellipsis}\):

• Divide the second term, -1, by the first term, 4:\text{\textbackslash} ( \frac{-1}{4} \text{\textbackslash})
• Divide the third term, \(\frac{1}{4}\), by the second term, -1: \text{\textbackslash} ( \frac{1}{4} \text{\textbackslash} \textbackslash) :

In this series, the common ratio, \(\frac{-1}{4}\), indicates that each term is multiplied by -0.25 to get the subsequent term. This helps identify the pattern of the series and was used in writing the series in summation notation.

The first term of a series is simply the initial number in the sequence. Identifying the first term is crucial as it serves as the base value from which the terms multiply by the common ratio. In the series \(4 - 1 + \text{\textbackslash} \frac{1}{4} - \frac{1}{16} +\text{\textellipsis}\) the first term is 4.
In general, the first term is denoted by '\(a\)' in the formula for a geometric series \text{\textbackslash} ( \text{\textbackslash sum}_{n=0}^{\text{\textbackslash upperbound}} a \text{\textbackslash} r^n \text{\textbackslash}).
This means that the common ratio and the first term together define the entire series, making them constants for each term's multiplication. Understanding this helps in correctly identifying and writing geometric series in a concise form.