Sum of series is a very basic knowledge point in mathematics, and it is also a common calculation problem in mathematics. In practical application, the summation of series is often used in statistics, finance, physics and other fields. This paper will introduce the basic methods and skills of series summation, including common series summation formulas and example analysis.

Basic concept of summation of sequence

A sequence is a series of numbers arranged according to certain rules. Sum of series is the process of adding these numbers. In mathematics, the sum of a series is usually represented by the symbol σ, the number after σ indicates the number of items to be summed in the series, the subscript indicates the starting item in the series, and the superscript indicates the ending item in the series. For example, σ n is the sum of all natural numbers from 1 to n.

Commonly used summation formula of sequence

The summation formula of 1. arithmetic progression.

Arithmetic progression is a series with equal difference between two adjacent terms in an exponential series. Arithmetic progression's summation formula is:

S=(a 1+an)×n/2

Where a 1 is the first item in arithmetic progression, an is the last item in arithmetic progression, and n is the number of items in arithmetic progression.

2. Sum formula of equal ratio series

Geometric series refers to the series in which the ratio of two adjacent terms in exponential series is equal. The summation formula of equal ratio series is:

s=a 1×( 1-q^n)/( 1-q)

Where a 1 is the first term of geometric series, q is the common ratio of geometric series, and n is the number of terms of geometric series.

3. Sum formula of square sequence

Square series refers to a series in which every term in an exponential series is the square of the previous term. The summation formula of square series is:

S=(2n^3+3n^2+n)/6

Where n is the number of terms in a square series.

An example analysis of summation of sequence

Here are a few examples to demonstrate the specific operation steps of sequence summation.

Example 1: Find the sum of natural numbers from 1 to 100.

Solution: According to the basic concept of sequence summation, we can express the sum of natural numbers from 1 to 100 as σ n, where the range of n is 1 to 100. According to arithmetic progression's summation formula, we can get:

s =(a 1+an)×n/2 =( 1+ 100)× 100/2 = 5050

Therefore, the sum of natural numbers from 1 to 100 is 5050.

Example 2: Find the sum of squares from 1 to 10.

Solution: According to the basic concept of series summation, we can express the sum of squares of 1 to 10 as σ σ σσn^2, where the value range of n is 1 to 10. According to the summation formula of square series, we can get:

s=(2n^3+3n^2+n)/6=(2× 10^3+3× 10^2+ 10)/6=385

Therefore, the sum of squares from 1 to 10 is 385.