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Euler summation formula
Euler's summation formula is a mathematical formula to solve the sum of arithmetic progression. It was put forward by the Swiss mathematician Euler in18th century, and can be used to calculate the sum of some items in arithmetic progression.

1. Definition and symbolic representation

Euler summation formula is used to calculate the sum of arithmetic progression. Arithmetic progression means that the difference between each item in the index column and the previous item is equal. The general form of Euler's summation formula is S=(n/2(a+l, where s represents the sum of arithmetic progression, n represents the number of terms, a represents the first term, and l represents the last term.

2. Derivation process

The derivation of Euler's summation formula can be obtained by adding arithmetic progression in reverse. Assuming that the first term of arithmetic progression is A, the tolerance is D, and the last term is L, the sum of each term is a+l, and * * * has n terms. Because the double sum contains the sum of the original sequence, it needs to be divided by 2, and finally the Euler sum formula is obtained.

3. Application

Euler summation formula is widely used in mathematics. It can be used to calculate the sum of a certain number of terms in arithmetic progression, thus simplifying the summation operation. Euler's summation formula can also be used to derive other mathematical formulas, such as arithmetic progression's general term formula and arithmetic progression's square sum formula. In addition, Euler summation formula can also be used to solve practical problems, such as calculating the sum of continuous variables such as time, distance and speed.

Step 4: Example

Examples are given to illustrate the application of Euler summation formula. Suppose there is a arithmetic progression, the first term is 3, and the tolerance is 2. Find the sum of the previous 10 items. According to Euler's summation formula, S=( 10/2(3+l=5(3+l)). Calculate the last term L = A+(n-1d = 3+9 (2 = 21) first, and substitute it into the formula to get S=5(3+2 1= 120). Therefore, the sum of the former items 10 is 120.

Generally speaking, Euler summation formula is a mathematical formula used to calculate arithmetic progression summation. It can be obtained by adding arithmetic progression in reverse, and the form of the formula can be obtained by deduction. Euler summation formula is widely used in mathematics, which can simplify summation operation, deduce other mathematical formulas and solve practical problems. Through Euler's summation formula, we can calculate the sum of arithmetic progression more conveniently.