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所在平台: Udemy |
课程主页: https://www.udemy.com/course/applied-mathematics-sequence-and-series/
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Coursera 应用数学：数列与级数 课程总结 本课程深入探讨了数列与级数的核心概念，涵盖了算术级数（A.P.）和几何级数（G.P.）及其相关性质。 * **数列与级数定义**: 课程首先阐述了数列的定义，即按照特定规则排列的数字序列，并指出数列可视为以自然数集为定义域的函数。有限数列称为有限数列，无限的则为无限数列。级数则是由数列各项相加构成的表达式，同样也存在有限级数和无限级数之分。 * **算术级数 (A.P.)**: 学习了算术级数的定义，即各项之间存在恒定公差的数列。课程介绍了算术级数的首项 (a)、公差 (d) 和末项 (l) 的表示方法，以及其通项公式 an = a + (n - 1) d。同时，详述了算术级数前 n 项和的两种计算公式：Sn = n/2 [2a + (n-1)d ] 和 Sn = n/2 (a + l)。 * **算术平均数 (A.M.)**: 讲解了两个数 a 和 b 的算术平均数的定义，即 (a + b) / 2，并指出 a, A, b 构成一个等差数列。 * **几何级数 (G.P.)**: 介绍了几何级数的定义，即任意项与其前一项之比恒定的数列，这个恒定比值称为公比。课程指出了几何级数首项 (a) 和公比 (r) 的表示。 * **几何平均数 (G.M.)**: 探讨了两个正数 a 和 b 的几何平均数，即通过构造 a, G, b 构成等比数列时的 G 值。 * **无限几何级数**: 课程最后涵盖了无限几何级数及其求和，虽然在概述中仅提到，但这是理解级数的重要部分。 * **A.M. 和 G.M. 的关系**: 课程还提及了算术平均数和几何平均数之间的关系，这是分析数列性质的重要工具。 本课程为学习者提供了扎实的数列与级数基础，对于理解更高级的数学和应用领域至关重要。
Sequence and SeriesSequence and SeriesArithmetic Progression (A.P.)Arithmetic Mean (A.M.)Geometric Progression (G.P.)General term of a G.P.Sum of n terms of a G.P.Arithmetic and Geometric series infinite G.P. and its sumGeometric mean (G.M.)Relation between A.M. and G.M.SUMMARY1. By a sequence, we mean an arrangement of number in definite order according to some rule. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3,....k}. A sequence containing a finite number of terms is called a finite sequence. A sequence is called infinite if it is not a finite sequence. 2. Let a1 , a2 , a3 ,.be the sequence, then the sum expressed as a1 + a2 + a3 +.is called series. A series is called finite series if it has got finite number of terms. ®3. An arithmetic progression (A.P.) is a sequence in which terms increase or decrease regularly by the same constant. This constant is called common difference of the A.P. Usually, we denote the first term of A.P. by a, the common difference by d and the last term by l. The general term or the n th term of the A.P. is given by an = a + (n - 1) d. The sum Sn of the first n terms of an A.P. is given by Sn = n/2 [2a + (n-1)d ] = n/2 (a + 1).4. The arithmetic mean A of any two numbers a and b is given by (a + b) / 2 i.e., the sequence a, A, b is in A.P. 5. A sequence is said to be a geometric progression or G.P., if the ratio of any term to its preceding term is same throughout. This constant factor is called the common ratio. Usually, we denote the first term of a G.P. by a and its common ratio by r.6. The geometric mean (G.M.) of any two positive numbers a and b is given by the sequence a, G, b is G.P.
