In a series, each number has a position, which we call an item. If every item in a series is arranged according to certain rules, then the series becomes a function. The function summation formula is used to calculate the sum of the items in this sequence.
The form of function summation formula can be expressed as:
S = a 1+a2+a3 +...+ an
Where s represents the sum of series, a 1, a2, a3, ..., representing each item in the series. This formula can be applied to various types of series, and the sum of series can be obtained simply by bringing each term into the formula.
Properties of function summation formula
The function summation formula has some important properties, which can help us to better understand and apply this formula. The following are some common attributes:
1. additivity: If a series can be divided into two parts, then the sum of this series can be expressed as the sum of these two parts. That is S = S 1+S2.
2. Subtraction property: If the sum of two series is the same, then their difference is also the same. That is S 1-S2 = S3-S4.
3. Linearity: If two series have the same coefficient k, then their sum also has the same coefficient k .. that is, k(S 1
+ S2) = kS 1 + kS2 .
4. Laurent series: If a function can be written as an infinite series, then this series is called Laurent series. In some mathematical problems, the function summation formula can be used to solve Laurent series.
5. Recursive formula: Some series can be solved by recursive formula, and the recursive formula can calculate the next item through each item in the series. Recursive formula can be transformed into function summation formula, which helps us to calculate the sum of series better.
How to apply function summation formula
When applying the function summation formula, it is necessary to understand some basic characteristics of the sequence, such as the first term, tolerance and number of terms of the sequence. Knowing these basic characteristics, we can bring each term into the function summation formula to solve it.
For example, we want to calculate the sum of the following series:
3,6,9, 12, 15, 18,2 1
The first term of this series is 3, the tolerance is 3 and the number of terms is 7. We can bring this information into the function summation formula and get:
s = 3+6+9+ 12+ 15+ 18+2 1
S = 3( 1 + 2 + 3 + 4 + 5 + 6 + 7)
S = 3(28)
S = 84
So the sum of this series is 84.
In practical application, the function summation formula is often used to calculate some continuous values, such as total income and total cost in a period of time. In addition, the function summation formula is often used to solve practical problems in statistics, physics, engineering and other fields.
conclusion
Function summation formula is a relatively basic knowledge point in mathematics, and it is also one of the widely used tools. Mastering this formula can help us better understand the sequence of numbers and solve practical problems. In practical application, we need to know the basic characteristics of sequence, and bring these characteristics into the function summation formula to solve.