Express each sum using summation notation. Use $1$ as the lower limit of summation and $i$ for the index of summation. $4+4^2/2+4^3/3+\cdots +4^n/n$

Verified step by step guidance

1. Identify the pattern in the given sum: the first term is \(4\), the second term is \(\frac{4^2}{2}\), the third term is \(\frac{4^3}{3}\), and so on, up to \(\frac{4^n}{n}\).
2. Notice that each term can be written as \(\frac{4^i}{i}\), where \(i\) is the index of summation starting from 1 and going up to \(n\).
3. Since the problem asks to express the sum using summation notation with lower limit 1 and index \(i\), write the sum as \(\sum_{i=1}^n \frac{4^i}{i} \).
4. This notation compactly represents the entire sum from the first term to the \(n\)-th term, capturing the pattern of powers of 4 divided by their respective indices.
5. Thus, the summation notation for the given sum is \(\sum_{i=1}^n \frac{4^i}{i} \).

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 (here, 1), and the upper limit (n) shows where it ends. Properly setting these limits is essential for accurately expressing the sum.

General Term of a Series

The general term defines the formula for each term in the sum based on the index. In this problem, each term is of the form 4^i divided by i. Identifying this pattern allows you to write the entire sum compactly using summation notation.