Infinite Series of Real Numbers

Abstract

The sums

$$\sum\limits_{k = 1}^n {{a_k} = {a_1} + \cdot \cdot \cdot + {a_n},\,\,\,\,\,\sum\limits_{k = 0}^n {{a_k} = {a_0} + \cdot \cdot \cdot + {a_n},} }$$

where n is some positive integer, were defined in Chapter II. They are examples of finite sums. Now we define the “sum” of the infinite series

$$\sum\limits_{n = 1}^\infty {{a_n} = {a_1} + {a_2} + \cdot \cdot \cdot \,\,\,or\,\,\,\,\,\sum\limits_{k = 0}^\infty {{a_k} = {a_0} + {a_1} + \cdot \cdot \cdot \,\,.} }$$