Problem 18 - Calculus: One and Several Variables

Question

Write the given sums as \(\sum_{k=3}^{10} a_{k}\) and as \(\sum_{i=0}^{7} a_{i+3}\) $$\frac{3^{3}}{3 !}+\frac{4^{4}}{4 !}+\cdots+\frac{10^{10}}{10 !}$$

Step-by-Step Solution

Answer

The series $$\frac{3^{3}}{3 !} + \frac{4^{4}}{4 !} + \cdots + \frac{10^{10}}{10 !}$$ can be represented as both \(\sum_{k=3}^{10} a_{k}\) and \(\sum_{i=0}^{7} a_{i+3}\), where \(a_k = \frac{k^k}{k!}\) and \(a_{i+3} = \frac{(i+3)^{(i+3)}}{(i+3)!}\).

Step 1: Analyze the pattern of the series

The terms in the sum series have a specific pattern: the base and exponent of the number is the same and this number is divided by its own factorial. This pattern can be described mathematically as \(\frac{k^k}{k!}\)

Step 2: Write the sum as \(\sum_{k=3}^{10} a_{k}\)

Replace the pattern of the series (which we previously defined as \(\frac{k^k}{k!}\)) with \(a_k\). Then, replace each term in the series with the corresponding \(a_k\), starting from \(k=3\) to \(k=10\). The sum can therefore be represented as \(\sum_{k=3}^{10} a_{k}\)

Step 3: Write the sum as \(\sum_{i=0}^{7} a_{i+3}\)

For this representation, we start from \(i=0\) and add 3 to \(i\) in our pattern definition to align it to the original series (since the original series starts at 3). So for each term in our original series, it is represented by \(a_{i+3}\) starting from \(i=0\) to \(i=7\). Therefore the series can also be represented as \(\sum_{i=0}^{7} a_{i+3}\)

Key Concepts

Series Representation

When dealing with sums or series in calculus, one way to efficiently express large additions is through series representation. This technique allows us to write lengthy sums in a compact mathematical notation, using sigma (\(\sum\)) notation. The notation captures the pattern of the series, defining both the starting and ending points of the terms being summed. For example, instead of writing each term separately in a sum from 3 to 10 of the expression \(\frac{k^k}{k!}\), we can express it as \(\sum_{k=3}^{10} \frac{k^k}{k!}\).

Series representation is particularly useful when a recognizable pattern or function can represent each term in the series. This not only simplifies the written expression but also allows us to perform operations and derive properties of the series without explicitly writing and calculating each individual term. By transforming a series into sigma notation, calculations become more efficient and manageable, especially for series spanning a large number of terms.

Factorials in Series

Factorials are a frequent mathematical operation used in sequences and series to express the product of all positive integers up to a certain number \(n\), represented as \(n!\). In series, factorials can appear in the denominator of terms, as seen with the sequence \(\frac{k^k}{k!}\). Here, the denominator \(k!\) rapidly grows as \(k\) increases, and acts as a normalizing factor that impacts the behavior and convergence of the series.

For example, consider the term \(\frac{4^4}{4!}\). Here, \(4! = 4 \times 3 \times 2 \times 1 = 24\), significantly reducing the value of \(4^4\) which is 256. This method of employing factorials keeps the growth of terms in check, which is critical in the analysis of series and their convergence properties.

• Factorials grow fast, influencing how quickly terms decrease in a series.
• They ensure that terms in the series don't soar to infinity, hence the series potentially converges.
• Factorials are pivotal in permutations and combinations.

Understanding the role of factorials is essential to mastering series representations and grasping how components within these expressions manipulate the overall behavior.

Mathematical Patterns

Mathematical patterns are the key to understanding complex series and sequences. A pattern is essentially a repeatable rule or structure within a series that can help identify and create formulas for computation. Recognizing and exploiting these patterns is crucial in simplifying complex mathematical problems.

In our example series \(\frac{k^k}{k!}\) from \(k=3\) to \(k=10\), the mathematical pattern is evident: each term's base and exponent match the current step \(k\), derived systematically from its position. This pattern helps in converting the verbose sum into the compact sigma notation.

Identifying patterns can turn arduous calculations into simple formulas, saving time and reducing errors. When you spot patterns:

• Look for repetitions or consistent transformations between terms.
• Note how patterns align with known mathematical functions.
• Utilize these observations to establish concise expressions.

By translating real-world problems into mathematical expressions through pattern recognition, we can harness the power of mathematics for solutions that would otherwise be cumbersome.