Problem 49 - College Algebra

Question

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\dots+\frac{14}{14+1}$$

Step-by-Step Solution

Answer

The series \(\frac{1}{2} + \frac{2}{3} + \frac{3}{4} + ... + \frac{14}{15}\) can be written using the summation notation as \( \sum_{i=1}^{14}\frac{i}{i+1}\).

Step 1: Identify the Pattern

For every term in this series, the numerator corresponds to the term's index number in the series (for instance, the term \(\frac{2}{3}\) is the second term, so the numerator is 2). In other words, for each term, the denominator is one greater than the numerator. Therefore, we can say that each term takes the form of \(\frac{i}{i+1}\), where 'i' is the term’s index number.

Step 2: Translate into Summation Notation

Given that our general term is \(\frac{i}{i+1}\), and we are given that the series starts with 1 and ends at 14 with 'i' as the index, we can write the series in summation notation. Thus, the series \(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{14}{15}\) translates to \( \sum_{i=1}^{14}\frac{i}{i+1}\).