- Start with the convergent geometric series formula \(\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}\) for \(|x|<1\).
- Apply term-by-term differentiation to both sides of the equation with respect to \(x\).
- Simplify the resulting expression to find that \(\sum_{n=1}^{\infty} n x^{n-1} = \frac{d}{dx} (\frac{1}{1-x}) = \frac{1}{(1-x)^2}\).
- Generalize the method to higher-order derivatives for more complex power series.
- Final Answer: \(\boxed{\frac{1}{(1-x)^2}}\)
Explanation
- Problem Analysis and Geometric Series Foundation
The objective is to find the sum of the power series \(\sum_{n=1}^{\infty} n x^{n-1}\) for \(|x|<1\). We begin with the well-known geometric series:
\[ \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \dots = \frac{1}{1-x} \]
This series converges for \(|x|<1\). The power series representation allows us to perform term-by-term differentiation within its interval of convergence.
- Derivation through Differentiation
To find the sum of the series \(\sum_{n=1}^{\infty} n x^{n-1}\), we differentiate both sides of the geometric series formula with respect to \(x\):
\[ \frac{d}{dx} \left( \sum_{n=0}^{\infty} x^n \right) = \frac{d}{dx} \left( \frac{1}{1-x} \right) \]
The derivative of the sum on the left side is:
\[ \frac{d}{dx} (1 + x + x^2 + x^3 + \dots) = 0 + 1 + 2x + 3x^2 + \dots = \sum_{n=1}^{\infty} n x^{n-1} \]
The derivative of the function on the right side using the chain rule is:
\[ \frac{d}{dx} ((1-x)^{-1}) = -1(1-x)^{-2} \cdot (-1) = \frac{1}{(1-x)^2} \]
Equating these results gives:
\[ \sum_{n=1}^{\infty} n x^{n-1} = \frac{1}{(1-x)^2}, \quad |x|<1 \]
This series converges for all \(|x|<1\).
- General Approach for Similar Series
For any series of the form \(\sum n a_n x^{n-1}\) where \(\sum a_n x^n\) is a known power series, we can find the sum by differentiating the sum function of the original series. By repeating this process (differentiating multiple times), one can find sums for series like \(\sum_{n=2}^{\infty} n(n-1) x^{n-2}\), which would involve the second derivative of the original sum function. The general approach remains the same: differentiate the closed-form sum function as many times as necessary to match the power of \(x\) and the coefficients in the series.
- Final Conclusion
The sum of the series \(\sum_{n=1}^{\infty} n x^{n-1}\) for \(|x|<1\) is \(\boxed{\frac{1}{(1-x)^2}}\).
Examples
Power series are fundamental in physics and engineering, particularly in the study of signals and systems. For example, if you are modeling the response of a filter, the behavior of the output can often be represented as a power series. Knowing how to manipulate these series, such as finding their sum through differentiation, allows engineers to derive closed-form expressions for transfer functions, which simplifies the analysis of system stability and efficiency. Understanding these operations is essential for mastering advanced mathematical modeling.