Question

The function defined by $$ f(x)=\sum_{n=0}^{\infty} c_{n} x^{n}, \quad \text { where } \quad c_{n+2}=\frac{2 n-\lambda}{(n+1)(n+2)} c_{n}, $$ can be written as \(f(x)=c_{0} g(x)+c_{1} h(x)\), where \(g\) is even and \(h\) is odd in \(x .\) Show that \(f(x)\) goes to infinity at least as fast as \(e^{x^{2}}\) does, i.e., \(\lim _{x \rightarrow \infty} f(x) e^{-x^{2}} \neq 0 .\) Hint: Consider \(g(x)\) and \(h(x)\) separately and show that $$ g(x)=\sum_{n=0}^{\infty} b_{n} x^{n}, \quad \text { where } \quad b_{n+1}=\frac{4 n-\lambda}{(2 n+1)(2 n+2)} b_{n} . $$ Then concentrate on the ratio \(g(x) / e^{x^{2}}\), where \(g\) and \(e^{x^{2}}\) are approximated by polynomials of very high degrees. Take the limit of this ratio as \(x \rightarrow \infty\), and use recursion relations for \(g\) and \(e^{x^{2}}\). The odd case follows similarly.

Short answer

The function \( f(x) \) can be proven to grow at least as fast as \( e^{x^{2}} \) at infinity, which means that the limit \( \lim_{x \rightarrow \infty} f(x) e^{-x^{2}} \) is not zero.

Step-by-step solution

Write the function \( g(x) \)

The function \( g(x) \) is given in a recursive form, we write it as \( g(x) = \sum_{n=0}^{\infty} b_{n} x^{2n} \), using the recursion relation \( b_{n+1}=\frac{4n-\lambda}{(2n+1)(2n+2)}b_{n} \). Now, we know that this series will represent an even function.

Simplify the ratio \( g(x) / e^{x^{2}} \)

Substituting the exponential series representation for \( e^{x^{2}} = \sum_{n=0}^{\infty} x^{2n} / n! \) into the ratio, we can simplify this to \( \sum_{n=0}^{\infty} b_{n} / n! \).

Take the limit of the ratio

To take the limit of this ratio, we need to consider the highest degree terms. By the recursion formula for \( b_{n} \), we know that \( b_{n} \) tends to a positive value for large \( n \). Thus, as \( x \) tends to infinity, the highest-degree terms will dominate and since it is a ratio of two exponential series, the limit will be a nonzero constant.

Consider the function \( h(x) \)

For the odd part of \( f(x) \), we can define \( h(x) = f(x) - c_{0} g(x) \), which will also follow a similar analysis, and hence will also tend to a nonzero constant at infinity. Thus \( f(x) \) grows at least as fast as \( e^{x^{2}} \) at infinity.

Key concepts

Series Expansion

Understanding series expansion is fundamental when exploring mathematical physics foundations. It involves expressing functions as sums of simpler, easier-to-manage terms. A typical form, as seen in the exercise, is the power series \( f(x) = \sum_{n=0}^{\infty} c_{n} x^{n} \), where each term \( c_{n} x^{n} \) represents a component of the function involving the variable \( x \) raised to some power \( n \).

Working with series expansions, you may notice patterns or 'recursion relations,' which help in defining the coefficients. This approach breaks down complex, potentially incomprehensible functions into more straightforward, calculable forms. The exercise requires students to explore these concepts by studying the recursive nature of the coefficients. In mathematical physics, series expansions like this are invaluable for approximating and comprehending behaviors of functions that cannot be solved explicitly.

Recursion Relations

Recursion relations in mathematical physics provide a systematic way of determining the coefficients of series expansions. In our example, the coefficient \( c_{n+2} \) is related to \( c_{n} \) by a specific formula, demonstrating how successive terms are connected. This recursive connection simplifies the process of computing an extensive series, as each next term depends on a known or previously computed term.

One improvement advice for students grappling with recursion relations is to look for patterns and ensure that the recurrence is applied accurately. Misapplication of these relations can lead to incorrect series and faulty conclusions. By focusing on them, students can derive other useful series representations or even identify asymptotic behaviors, which brings us to the next critical concept.

Asymptotic Behavior

Grasping Asymptotic Behavior

The asymptotic behavior of functions reveals how they behave as the variable approaches a particular value, often at infinity or zero. In our exercise, we're examining the function's growth compared to \( e^{x^2} \) as \( x \) goes to infinity. This function, representing the span of an endlessly increasing series, is essential for visualizing the complete nature of mathematical behaviors.

For better insight, students can approximate \( g(x) \) and \( e^{x^2} \) with polynomials of high degrees and analyze the ratio's limit as \( x \) tends towards infinity. The non-zero limit speaks volumes – it confirms that the original function \( f(x) \) indeed grows at least as fast as \( e^{x^2} \) does. This is a complex yet crucial aspect of mathematical physics that students should understand for analyzing functions at extreme scales.