Question

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.

The sum of a convergent series

Short answer

A series converges if its sequence of partial sums converges, and in that case, we define the sum of the series to be the limit of its partial sums. For example, $\sum _{n=1}^{\infty }\frac{1}{2^n}$ is the series which converges.

Step-by-step solution

Step 1. Given Information   

The given statement is the sum of a convergent series

Step 2. Explanation     

A series converges if its sequence of partial sums converges, and in that case, we define the sum of the series to be the limit of its partial sums.

The sum of the first terms of a series up to a point is another sequence called the partial sum. A series is convergent if its partial sum has a limit. We can say that a convergent series is equal to some finite value.

For example, $\sum _{n=1}^{\infty }\frac{1}{2^n}$is a convergent series as the partial sums will look like $\frac{1}{2},\frac{3}{4},\frac{7}{8},..$and converges to 1.