Problem 21 - Calculus

Question

Write in sigma notation. (a) \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{78}\) (b) \(x^{2}+2 x^{4}+3 x^{6}+4 x^{8}+\cdots+50 x^{100}\)

Step-by-Step Solution

Answer

(a) \(\sum_{n=2}^{78} \frac{1}{n}\); (b) \(\sum_{n=1}^{50} n \cdot x^{2n}\).

Step 1: Identify the sequence pattern for part (a)

The sequence given in part (a) starts at \( \frac{1}{2} \) and increases by adding fractions with denominators increasing by 1 incrementally, up to \( \frac{1}{78} \). This suggests a harmonic series where the general term is \( \frac{1}{n} \).

Step 2: Write sigma notation for part (a)

Since the sequence begins at the second term of the harmonic series (\( \frac{1}{2} \)) and ends at the 78th term (\( \frac{1}{78} \)), the sigma notation is: \[\sum_{n=2}^{78} \frac{1}{n}\]

Step 1: Identify the sequence pattern for part (b)

In part (b), each term of the sequence is a product of two factors: a linearly increasing coefficient and a power of \(x\) that increases by powers of 2. Specifically, terms are in the form \( n x^{2n} \), beginning with \( n=1 \) and ending with \( n=50 \).

Step 2: Write sigma notation for part (b)

The sequence given in part (b) can be expressed as a sum where each term follows the pattern \( n \cdot x^{2n} \). Hence, in sigma notation, it is: \[\sum_{n=1}^{50} n \cdot x^{2n}\]

Key Concepts

Harmonic Series

The harmonic series is a fascinating sequence in mathematics that involves the sum of reciprocals of natural numbers. A simple way to understand the harmonic series is to think about adding fractions whose denominators are increasing natural numbers. For example, the traditional harmonic series begins like this: \( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots \). In the given exercise, we look at a part of this series starting from \(\frac{1}{2}\) to \(\frac{1}{78}\).

The general term of a harmonic series can be represented as \( \frac{1}{n} \), where \( n \) is a natural number. This means each term is simply the reciprocal of a natural number. Although the individual terms become smaller as \( n \) increases, the series itself grows without bound, leading to the intriguing result that the harmonic series diverges. In plain terms, if you keep adding more and more terms, the total sum will keep increasing indefinitely. This is a key characteristic that makes the harmonic series unique and widely studied in mathematical analysis.

Sequence Patterns

Recognizing sequence patterns is crucial for understanding how to express a sequence in sigma notation. In mathematics, a sequence is simply an ordered list of numbers following a particular rule. In the exercise, we are given two different sequence patterns that we need to express in sigma notation.

For part (a) of the exercise, the sequence is part of a harmonic series where each term is a reciprocal of consecutive integers starting from 2 up to 78. Identifying this pattern allows us to write the series in compact sigma notation, making it easier to work with.

In part (b), the sequence pattern becomes slightly more complex. Each term is formed by multiplying a linearly increasing coefficient \( n \) with a power of \( x \) that increases exponentially by 2. The general term here is \( n \cdot x^{2n} \). Recognizing the pattern of multiplication and exponential growth is essential for writing this expression using summation notation.

Summation Notation

Summation notation, often referred to as sigma notation, is a helpful shorthand way of expressing the sum of a sequence of numbers. Instead of writing out every individual term, we use the Greek letter sigma (\( \Sigma \)) to signify summation.

The general form of sigma notation is \( \sum_{n=a}^{b} f(n) \), where \( f(n) \) is the function representing each term in the sequence, and \( a \) and \( b \) are the lower and upper limits of summation, specifying where the sequence starts and ends. This makes it easier to manage and calculate sums over potentially large numbers.

In our exercise, the sequence from \( \frac{1}{2} \) to \( \frac{1}{78} \) is rewritten as \( \sum_{n=2}^{78} \frac{1}{n} \) using sigma notation. Similarly, the more complex sequence in part (b) follows a specific pattern \( n \cdot x^{2n} \) and is expressed as \( \sum_{n=1}^{50} n \cdot x^{2n} \). Understanding and using summation notation allows us to grasp the structure of sequences better and provides a powerful tool for solving complex summations efficiently.