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 sum: each term has the form \( \frac{i}{i+1} \), where \( i \) starts at 1 and increases by 1 for each term.
2. Determine the number of terms in the sum. Since the last term is \( \frac{14}{14+1} \), the index \( i \) goes from 1 to 14.
3. Write the summation notation using the index \( i \), the lower limit 1, and the upper limit 14, with the general term \( \frac{i}{i+1} \).
4. Express the sum as: \[ \sum_{i=1}^{14} \frac{i}{i+1} \]
5. This notation compactly represents the entire sum \( \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \cdots + \frac{14}{15} \).

Key Concepts

Summation Notation

Summation notation is a concise way to represent the sum of a sequence of terms using the sigma symbol (∑). It includes an index of summation, lower and upper limits, and a general term formula. This notation simplifies writing long sums and helps in analyzing series.

Index of Summation and Limits

The index of summation (commonly i) represents the variable that changes in each term of the sum. The lower limit indicates where the summation starts, and the upper limit shows where it ends. Correctly identifying these limits is essential to accurately express the sum.

General Term of the Sequence

The general term defines the formula for each term in the sum based on the index i. For the given sum, each term is a fraction with numerator i and denominator i+1. Recognizing this pattern allows writing the sum compactly using summation notation.