Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(\sum_{i=1}^{n} \frac{4^{i}}{i}\)

Step 1: Identify the pattern

Look at the pattern in the given sequence. The numerator is 4 raised to the power of n, while the denominator is the number corresponding to each term. This indicates that the exponent and the denominator both rely on the position within the series.

Step 2: Represent in summation notation

Once the pattern is identified, represent it using sigma notation. It is clearly observed that the series starts from 1 and goes to n (terms). Thus, in sigma notation, let's say that the index of summation is 'i' which can take values from 1 to 'n'. The general term is \(\frac{4^{i}}{i}\). So, put this all together to form \(\sum_{i=1}^{n} \frac{4^{i}}{i}\) as the sigma notation for the given series.

Key Concepts

Sigma Notation

Sigma notation is a way of writing a sum of multiple terms in a concise form. It's heavily used in mathematics, particularly in calculus and statistics, to represent long sums without writing every term individually. The Greek letter \(\Sigma\) is used to denote summation. It's akin to saying "add these terms up" without actually listing them all.

In the given problem, the goal is to express a series in summation notation. The formula is \(\sum_{i=1}^{n} \frac{4^i}{i}\),where:

• \(i\) is the index of summation.
• The lower limit is 1, indicating where the series starts.
• \(n\) is the upper limit, showing where the series ends.

This compact expression replaces what could otherwise be a long addition of fractions with powers of 4 in the numerators and consecutive numbers in the denominators.

Series Pattern

Understanding the pattern in a series is crucial for writing it in summation notation. A series is essentially a sequence of numbers added together. The series pattern helps us understand how each term in the sequence is constructed.

For the sequence given in the exercise, each term takes the form of \(\frac{4^n}{n}\). This suggests a clear pattern: each term's power and denominator increase together. The numerator of each fraction follows the pattern \(4^i\), where the base, 4, remains constant, and the exponent, \(i\), increases incrementally.

The denominator mirrors the position of the term within the sequence, meaning it's \(i\) as well. By observing this pattern, you can generalize this sum efficiently using sigma notation.

Sequence Representation

Sequence representation refers to the way each element of a series is expressed as a function of its position. It's the mathematical blueprint of the series. In order to express a sequence correctly in summation notation, you need to figure out how to represent each term based on its position.

Let's break down the sequence from our exercise. It begins with 4, followed by \(\frac{4^2}{2}\), then \(\frac{4^3}{3}\), continuing in this manner up to \(\frac{4^n}{n}\). Each term can be understood by looking at its position \(i\):

• When \(i = 1\), the term is \(\frac{4^1}{1} = 4\).
• When \(i = 2\), the term is \(\frac{4^2}{2} = \frac{16}{2}\).
• When \(i = 3\), the term is \(\frac{4^3}{3} = \frac{64}{3}\).

The sequence elements are consistent with raising 4 to the \(i\)-th power, divided by \(i\) itself.

Index of Summation

The index of summation plays a key role in sigma notation; it represents a variable part of the formula used to build each term in the series. Commonly denoted by letters like \(i, j, k,\) etc., it iterates from the lower limit to the upper limit within the summation expression.

The index determines which term in the sequence you're currently evaluating, essentially acting as a placeholder for the term position. In our example, the index \(i\) begins at 1, increases by 1 with each term added, and concludes at \(n\).

• This lets you systematically construct each term: \( \frac{4^i}{i} \).
• Starting from \(i=1\) through \(i=n\), the expression \(\sum_{i=1}^{n} \frac{4^i}{i}\) succinctly summarizes the entire series.

Understanding and manipulating the index of summation is fundamental when working with series or performing complex summations.