Problem 59 - Single Variable Calculus

Question

Find the sum of the given series. (Hint: Each series is the Maclaurin series of a function evaluated at an appropriate point.) \(\sum_{n=0}^{\infty}(-1)^{n} \frac{\pi^{2 n}}{(2 n) !}\)

Step-by-Step Solution

Answer

The sum of the given series is \(\cos(\pi) = -1\).

Step 1: Recognize function's Maclaurin series

The given series resembles the Maclaurin series of the cosine function: \[f(x) = \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\] Now, we need to find a point at which this cosine function, if applied, will give us our original series.

Step 2: Find the appropriate point

To transform the cosine function's Maclaurin series into our given series, we need to replace the variable x with \(\pi\). This will make the series become: \[\cos(\pi) = \sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n}}{(2n)!}\] Which is the sum of the given series.

Step 3: Calculate the sum

Now that we know the given series is equivalent to the Maclaurin series of the cosine function at \(\pi\), we can find the sum of the series by simply evaluating the cosine function at that point: \[\sum_{n=0}^{\infty}(-1)^{n} \frac{\pi^{2 n}}{(2 n) !} = \cos(\pi) = -1\] Thus, the sum of the given series is -1.

Key Concepts

Convergence of Series

Understanding the convergence of a series is crucial when dealing with infinite sums. A series is simply a sum of terms of a sequence. However, when a series is infinite, the question arises whether it approaches a specific value or not. This specific value, if it exists, is called the limit of the series. When a series has a limit, we say that it converges; if not, it diverges.

In the context of the Maclaurin series for the cosine function, the convergence is determined by the ratio test, which shows that it converges for all real numbers. This is because the factorial in the denominator grows much faster than the power of the numerator, making the terms of the series shrink rapidly as n increases. Hence, the sum of the series becomes stable at a certain value.

Cosine Function

The cosine function is one of the primary trigonometric functions, relating the angle of a right triangle to the ratio of the adjacent side over the hypotenuse. In calculus and series expansion, the cosine function can be represented as an infinite series known as the Maclaurin series, which is a type of Taylor series centered at zero.

The Maclaurin series of the cosine function is expressed as:\[\begin{equation}\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}\end{equation}\]This elegant formula allows us to evaluate cosine values for all real numbers using an infinite series, which converges and accurately represents the cosine function for all inputs. It's particularly useful in contexts where computing trigonometric functions directly is complex or impossible.

Factorial Notation

Factorial notation is essential for understanding series convergence and evaluating mathematical expressions involving permutations or combinations. The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. For instance, 3! is 3 x 2 x 1, which equals 6. Importantly, by definition, 0! is equal to 1.

Factorials grow at an incredibly fast rate with larger values of n. This rapid growth impacts the convergence of series, particularly those where factorial terms are present in the denominator, like the Maclaurin series of the cosine function, contributing to the series' convergence for all real numbers.

Series and Sequence Calculus

Series and sequence calculus is the study of sequences (ordered lists of numbers following a specific rule) and series (the sum of the elements of a sequence). In calculus, series can represent functions, such as in the case of the Taylor or Maclaurin series, which can approximate calculus functions to a high degree of accuracy.

To work with series and sequence calculus effectively, it's vital to understand various concepts like convergence (as previously explained), operations on series, and the role of sequences in defining series. Each series can be used to express more complex mathematical functions, and in the context of the exercise, the Maclaurin series is used to represent the cosine function to find the sum.