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One of the core concepts in summation is the ability to split sums into smaller, more manageable parts. This technique makes it easier to solve complex series by breaking them down into simpler components.
For example, in the summation \(\textstyle \sum_{i=1}^{5}(4i^2 - 2i + 6)\), we can split it into three separate sums:
\[\sum_{i=1}^{5}(4i^2 - 2i + 6) = \sum_{i=1}^{5}4i^2 - \sum_{i=1}^{5}2i + \sum_{i=1}^{5}6\]
This breaks the original complex series into three simpler series: \(\textstyle \sum_{i=1}^{5}4i^2\), \(\textstyle \sum_{i=1}^{5}2i\), and \(\textstyle \sum_{i=1}^{5}6\).
Splitting sums is useful because it allows you to apply different summation properties and rules to each part separately.

Summation formulas are predetermined equations that simplify the process of evaluating series. By knowing a few key formulas, you can quickly find the sum of many types of sequences:

• The sum of the first n natural numbers: \(\textstyle \sum_{i=1}^{n}i = \frac{n(n+1)}{2}\)
• The sum of the squares of the first n natural numbers: \(\textstyle \sum_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6}\)
• The sum of any constant c added n times: \(\textstyle \sum_{i=1}^{n}c = n\cdot c\)

For example, to evaluate \(\textstyle \sum_{i=1}^{5}i\), we use the first formula:
\[\sum_{i=1}^{5}i = \frac{5(5+1)}{2} = 15\]
Similarly, for \(\textstyle \sum_{i=1}^{5}i^2\), we use the second formula:
\[\sum_{i=1}^{5}i^2 = \frac{5(5+1)(2 \times 5+1)}{6} = 55\]
Using these formulas makes the computation straightforward and quick.

Once you've split your sums and applied the appropriate summation formulas, the next step is to evaluate the series. This involves substituting back the computed sums and performing basic arithmetic.
For our example series \(\textstyle \sum_{i=1}^{5}(4i^2 - 2i + 6)\):

• We first evaluated \(\textstyle \sum_{i=1}^{5}4i^2 = 4 \times 55 = 220\)
• Then \(\textstyle \sum_{i=1}^{5}2i = 2 \times 15 = 30\)
• And \(\textstyle \sum_{i=1}^{5}6 = 6 \times 5 = 30\)

Putting these values back into the original series, we get:
\[4 \sum_{i=1}^{5}i^2 - 2 \sum_{i=1}^{5}i + 6 \sum_{i=1}^{5}1 = 220 - 30 + 30 = 220\]
You've now successfully evaluated the series. Knowing these methods and formulas will make tackling any series much more manageable.