Problem 17 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Consider the sum $$ q^{5}-q^{7}+q^{9}-q^{11}+\cdots+q^{41} $$ (a) Put the sum into closed form. (b) Put the sum into summation notation. (c) Now put \(-q^{5}+q^{7}-q^{9}+q^{11}-\cdots-q^{41}\) into summation notation.

Step-by-Step Solution

Answer

(a) The closed form of the sum is \(S = q^5 (1 - (-q^2)^{19} ) / (1-(-q^2))\). (b) The summation notation of the sum is \(\sum_{n=1}^{19} -1^{n-1}q^{2n+3}\). (c) The summation notation of the sum \(-q^{5}+q^{7}-q^{9}+q^{11}-\cdots-q^{41}\) is \(\sum_{n=1}^{19} -1^n q^{2n+3}\).

Step 1: Identify the pattern

Firstly, identify the pattern in the given series: \(q^{5}-q^{7}+q^{9}-q^{11}+\cdots+q^{41}\). Here, the powers of \(q\) increase by 2 for each term and the sign alternates starting with a positive sign.

Step 2: Find the closed form

The common ratio for this series is \(r = -q^2\), and the first term \(a = q^5\). The series is finite and ends at power 41, i.e., there are total of 19 terms. The sum of the series \(S_n = a(1 - r^n) / (1 - r)\). Substituting the values, we get \(S = q^5(1 - (-q^2)^{19} ) / (1 - (-q^2))\).

Step 3: Putting the series into summation notation

Summation notation is a convenient way to concisely represent the series. In this case, the series can be defined as \(-1^{n-1}q^{2n+3}\) for \(n = 1\) to \(n = 19\), which can be represented in sigma notation as \(\sum_{n=1}^{19} -1^{n-1}q^{2n+3}\).

Step 4: Putting the series into summation notation with reversed signs

The series \(-q^{5}+q^{7}-q^{9}+q^{11}-\cdots-q^{41}\) is similar to the original series, but starting with a negative sign. The summation representation will be \(-1^n q^{2n+3}\) for \(n = 1\) to \(n = 19\), or in sigma notation \(\sum_{n=1}^{19} -1^n q^{2n+3}\).

Key Concepts

Closed Form Expression

A closed form expression is a way of representing a series or a sequence in a simplified algebraic formula. This form provides a way to calculate any term in the series without writing out all the previous terms. For the series given in the exercise, \[ q^{5} - q^{7} + q^{9} - q^{11} + \cdots + q^{41} \]we can identify the first term as \( a = q^5 \) and the common ratio \( r = -q^2 \). A key feature of the closed form is that it provides a single expression to compute the sum of the series, which is especially useful for long series. Understanding the closed form for a geometric series like this involves spotting the first term and the common ratio, even when the series includes multiple elements, such as alternating signs. Applying the closed form summation formula for a finite geometric series, we find the expression \[ S = q^5 \frac{1 - (-q^2)^{19}}{1 - (-q^2)} \].

Summation Notation

Summation notation is a mathematical shorthand used to express a long sum using the Greek letter sigma (\( \Sigma \)). It's a compact way to denote series, so it's easier to interpret complex expressions without losing track. In the series \[ q^{5} - q^{7} + q^{9} - q^{11} + \cdots + q^{41} \],the alternating powers of \( q \) can be expressed concisely using summation notation. This sums up to:\[ \sum_{n=1}^{19} (-1)^{n-1} q^{2n+3} \]. This notation specifies that you start at \( n=1 \) and increase \( n \) up to 19. The powers of \( q \) depend on \( n \), identified by the formula \( 2n + 3 \), while the sign alternates based on \( (-1)^{n-1} \). Summation notation helps to succinctly communicate the idea of a sum, clearly indicating its range and pattern.

Alternating Series

An alternating series is a series whose terms alternate in sign, such as positive, negative, positive, and so on. This is a classic mathematical concept that arises in many contexts. In the series from the exercise, \[ q^{5} - q^{7} + q^{9} - q^{11} + \cdots + q^{41} \],the alternation is evident from the coefficients that switch signs at each step. This feature is elegantly captured in both closed forms and summation notation. Understanding alternating series involves recognizing the regularity of sign change, often represented in formulas using powers of \( -1 \). For the summation notation representations,\[ (-1)^{n-1} \] captures positive terms when \( n \) is odd, and negative terms when \( n \) is even.Such series can often have special convergence properties and are an important type of series in mathematical analysis.

Exponential Series

Exponential series involve terms that grow exponentially, which means each term is a power of a fixed base, such as in \[ q^5, q^7, q^9, \ldots, q^{41} \].These powers of \( q \) are governed by the base \( q \) and exponent rules. The defining element here is the regular pattern in the exponents, which increase by 2 in each term of the given series.Exponential patterns in series are often used to model real-world phenomena. They are defined by their growth potential, as each subsequent term increases rapidly compared to linear or polynomial patterns. This results in compact expressions in closed form and summation notation, making them powerful tools in calculations and proofs.Thus, in the summation notation, the pattern is clearly expressed as \( q^{2n+3} \), aligning with exponential growth rules, where exponent changes are predictable and structured.