In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)

Verified step by step guidance

1. Identify the pattern in the sequence: \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots, \frac{14}{15} \).
2. Notice that each term in the sequence can be expressed as \( \frac{i}{i+1} \), where \( i \) is the index of summation.
3. Determine the range of \( i \). The sequence starts at \( i = 1 \) and ends at \( i = 14 \).
4. Express the sum using summation notation: \( \sum_{i=1}^{14} \frac{i}{i+1} \).
5. Verify that the expression \( \sum_{i=1}^{14} \frac{i}{i+1} \) correctly represents the given sequence.

Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of terms. It typically uses the Greek letter sigma (Σ) to denote the sum, with an index of summation that indicates the starting and ending values. For example, Σ from i=1 to n of a_i represents the sum of the terms a_1, a_2, ..., a_n.

Index of Summation

The index of summation is a variable that represents the position of each term in the sequence being summed. In this case, 'i' is used as the index, starting from 1 and increasing by 1 for each subsequent term. It allows for the systematic representation of each term in the sum, making it easier to express complex series succinctly.

Sequence and Series

A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence. In the given problem, the sequence consists of fractions where the numerator is the index 'i' and the denominator is 'i+1'. Understanding how to identify the pattern in the sequence is crucial for correctly expressing the series in summation notation.