Problem 41 - Calculus: One and Several Variables

Question

Sum the series. $$\sum_{k=9}^{\infty} \frac{1}{k !} x^{3 k}$$

Step-by-Step Solution

Answer

The sum of the series is given by: \(\sum_{k=9}^{\infty} \frac{1}{k !} x^{3 k} = \frac{x^{27}}{9!(1-x^3)}\).

Step 1: Rewrite the series as a geometric series expression

We can rewrite the given series as: $$\sum_{k=9}^{\infty} \frac{1}{k !} x^{3 k}$$ $$= x^{27} \sum_{k=9}^{\infty} \frac{1}{k !} x^{3 (k-9)}$$ Now, let's define a new variable \(n\) such that \(n = k - 9\): $$= x^{27} \sum_{n=0}^{\infty} \frac{1}{(n+9)!} x^{3n}$$

Step 2: Recognize the form of the sum and simplify

The sum in the series now has the form of a standard geometric series: $$\sum_{n=0}^{\infty} ar^n$$ where \(a = \frac{1}{9!}\) and \(r = x^3\). We can now find the sum of the infinite geometric series using the formula: $$S = \frac{a}{1 - r}$$ Plug in the values for \(a\) and \(r\): $$S = \frac{\frac{1}{9!}}{1 - x^3}$$

Step 3: Consider the constant term multiplied by the sum

Finally, we must account for the constant term that we factored out at the beginning, \(x^{27}\): $$\sum_{k=9}^{\infty} \frac{1}{k !} x^{3 k} = x^{27} \cdot \frac{\frac{1}{9!}}{1 - x^3}$$

Step 4: Simplify the expression

Now, we can simplify the expression to find the sum of the series: $$\sum_{k=9}^{\infty} \frac{1}{k !} x^{3 k} = \frac{x^{27}}{9!(1-x^3)}$$

Key Concepts

Infinite Series

An infinite series is a sequence of numbers or expressions added together endlessly, continuing indefinitely. In mathematical terms, an infinite series is given by \[ \sum_{k=n}^{\infty} a_k \]where each term has a specific pattern or formula. The most fascinating feature of infinite series is whether they "converge" or "diverge."

• Convergence means that as we add more terms, the series approaches a specific value.
• Divergence means the series has no finite limit and continues to increase or decrease indefinitely.

In the exercise, the series is expressed in terms of factorials and powers: \( \sum_{k=9}^{\infty} \frac{1}{k!} x^{3k} \). We worked to convert it into a more recognizable infinite geometric series format, which allows us to easily evaluate its sum.

Factorial Notation

Factorial notation is symbolized by the exclamation point \(!\). It represents the product of all positive integers up to a certain number. For example:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)

Factorials frequently appear in infinite series, especially when terms involve permutations or combinations. In our series, factorial notation \( (k!) \) serves a critical role in influencing the behavior of the series. Each term's denominator becomes very large, contributing to convergence. The series transforms into a manageable form using the factorial in the denominator, assisting in calculating the infinite series accurately.

Power Series

A power series is a series of the form \[ \sum_{n=0}^{\infty} a_n (x-c)^n \]where:

• \( a_n \) are coefficients.
• \( x \) is a variable and \( c \) is a constant around which the series is centered.

Power series are incredibly versatile in calculus and mathematical analysis, allowing us to express complex functions in a series form, much like a polynomial. The exercise involves the conversion of the series into a geometric series which, in a broader sense, shares similarities with power series by utilizing powers of \( x \). By examining \( \sum_{n=0}^{\infty} \frac{1}{(n+9)!} x^{3n} \), we've utilized a strategic approach in aligning the structure with a geometric concept, simplifying summation.

Calculus Series Summation

Calculus includes powerful tools for summing series, especially when direct computation seems daunting. We've tackled a geometric form in our problem that applies sum formulas like: \[ S = \frac{a}{1-r} \]to efficiently determine series' values. Here's why such formulas are essential:

• The formula requires the common ratio \( r \) to be less than 1 for convergence.
• It simplifies an endless addition into a tangible figure.

The problem at hand simplified the infinite series with factorial and power expressions into a geometric series context. Recognizing this enabled us to use the calculus summing formula effectively, leading to the series' simple, elegant sum: \( \frac{x^{27}}{9!(1-x^3)} \). Through calculus, what started as an intimidating infinite series becomes manageable.