Solving the Infinite Series ( sum_{n1}^{infty} frac{n^3}{n!} ) Using the Exponential Function

The problem of solving the infinite series ( sum_{n1}^{infty} frac{n^3}{n!} ) involves several key steps, primarily relying on the properties of the exponential function and its derivatives. In this article, we'll walk through each step in detail, providing a comprehensive explanation of the methodology and final result.

Understanding the Series

The given series is:

[ sum_{n1}^{infty} frac{n^3}{n!} ]

To approach this problem, we start by recalling the Taylor series expansion for the exponential function ( e^x ).

Recall the Taylor Series for ( e^x )

The Taylor series for ( e^x ) is expressed as:

[ e^x sum_{n0}^{infty} frac{x^n}{n!} ]

Deriving the Series Using ( e^x )

We can differentiate the Taylor series for ( e^x ) to generate terms involving ( n ). The first few derivatives are as follows:

First Derivative

The first derivative of ( e^x ) is:

[ frac{d}{dx} e^x e^x sum_{n1}^{infty} frac{n x^{n-1}}{n!} ]

Multiplying both sides by ( x ):

[ x e^x sum_{n1}^{infty} frac{n x^n}{n!} ]

Second Derivative

The second derivative of ( e^x ) is:

[ frac{d^2}{dx^2} e^x e^x sum_{n2}^{infty} frac{n(n-1) x^{n-2}}{n!} ]

Multiplying by ( x^2 ):

[ x^2 e^x sum_{n2}^{infty} frac{n(n-1) x^n}{n!} ]

Third Derivative

The third derivative of ( e^x ) is:

[ frac{d^3}{dx^3} e^x e^x sum_{n3}^{infty} frac{n(n-1)(n-2) x^{n-3}}{n!} ]

Multiplying by ( x^3 ):

[ x^3 e^x sum_{n3}^{infty} frac{n(n-1)(n-2) x^n}{n!} ]

Combining the Derivatives

To express ( n^3 ) in terms of ( n(n-1)(n-2) ), we observe that:

[ n^3 n(n-1)(n-2) 3n(n-1) n ]

Thus, we can write the series as:

[ sum_{n1}^{infty} frac{n^3}{n!} sum_{n1}^{infty} frac{n(n-1)(n-2)}{n!} 3 sum_{n1}^{infty} frac{n(n-1)}{n!} sum_{n1}^{infty} frac{n}{n!} ]

Calculating Each Term

Let's break down each term:

First Term

[ sum_{n3}^{infty} frac{n(n-1)(n-2)}{n!} sum_{n3}^{infty} frac{1}{(n-3)!} e ]

Second Term

[ 3 sum_{n2}^{infty} frac{n(n-1)}{n!} 3 sum_{n2}^{infty} frac{1}{(n-2)!} 3e ]

Third Term

[ sum_{n1}^{infty} frac{n}{n!} sum_{n1}^{infty} frac{1}{(n-1)!} e ]

Combining the Results

Putting it all together:

[ sum_{n1}^{infty} frac{n^3}{n!} e 3e e 5e ]

Final Result

Therefore, the sum is:

[ sum_{n1}^{infty} frac{n^3}{n!} 5e ]

[ text{Final Result: } sum_{n1}^{infty} frac{n^3}{n!} 5e ]

This approach showcases the power of the exponential function in solving complex series. The steps involved are fundamental in understanding the behavior of infinite series and their manipulation using derivatives.

Keywords: Exponential Function, Infinite Series, Taylor Series Expansion