Problem 60 - Thomas Calculus

Question

The sum of the series \(\sum_{n=0}^{\infty}\left(n^{2} / 2^{n}\right)\) . To find the sum of this series, express 1\(/(1-x)\) as a geometric series, differentiate both sides of the resulting equation with respect to \(x,\) multiply both sides of the result by \(x\) , differentiate again, multiply by \(x\) again, and set \(x\) equal to 1\(/ 2 .\) What do you get?

Step-by-Step Solution

Answer

The sum of the series is 8.

Step 1: Geometric Series Expression

We start with the geometric series formula: \[\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n\] This is valid for \(|x| < 1\).

Step 2: First Differentiation

Differentiate both sides with respect to \(x\):\[\frac{d}{dx}\left(\frac{1}{1-x}\right) = \sum_{n=0}^{\infty} \frac{d}{dx} x^n\]This gives us:\[\frac{1}{(1-x)^2} = \sum_{n=1}^{\infty} n x^{n-1}\]

Step 3: First Multiplication by x

Now, multiply both sides of the equation by \(x\):\[x \cdot \frac{1}{(1-x)^2} = x \cdot \sum_{n=1}^{\infty} n x^{n-1} = \sum_{n=1}^{\infty} n x^n\]

Step 4: Second Differentiation

Differentiate the equation obtained in the previous step:\[\text{Differentiate } \left( x \cdot \frac{1}{(1-x)^2} \right): \frac{d}{dx}\left(\frac{x}{(1-x)^2}\right) = \sum_{n=1}^{\infty} n^2 x^{n-1}\]Calculate the derivative:\[\frac{1 + x}{(1-x)^3}\]

Step 5: Second Multiplication by x

Multiply both sides by \(x\) one more time:\[x \cdot \frac{1 + x}{(1-x)^3} = \sum_{n=1}^{\infty} n^2 x^n\]

Step 6: Substitute x with 1/2

Finally, substitute \(x = \frac{1}{2}\) into the series:\[\frac{\frac{1}{2} + \frac{1}{2}}{(1-\frac{1}{2})^3} = \sum_{n=1}^{\infty} n^2 \cdot \left(\frac{1}{2}\right)^n\]Simplify:\[\frac{1}{(\frac{1}{2})^3} = \frac{1}{\frac{1}{8}} = 8\]

Key Concepts

Geometric Series

A geometric series is a series with constant ratio between successive terms. It starts with a simple formula that is incredibly useful:

• The geometric series formula is expressed as: \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \).

This means that any series of the form \( x^0 + x^1 + x^2 + \ldots + x^n \) can be simplified using this formula, as long as \(|x| < 1\). For our exercise, it serves as the starting point to break down and simplify the given series \( \sum_{n=0}^{\infty}\left(n^{2} / 2^{n}\right) \). The geometric series formula helps transform complex expressions into manageable ones, underscoring its importance in both elementary and advanced mathematics. Understanding how to manipulate this formula is key to dealing with more advanced series.

Differentiation

Differentiation is a fundamental tool in calculus used to find the rate at which a function is changing. In the context of series, differentiation helps us transform the expression to reveal patterns or simplify terms:

• First differentiation of the geometric series gives: \( \sum_{n=1}^{\infty} n x^{n-1} \), showing a more complex structure of the series.
• Once multiplied by \( x \), differentiating again helps in finding a power series to match the desired form \( \sum_{n=1}^{\infty} n^2 x^n \).

This process involves careful application of the power rule from calculus. The second differentiation, followed by algebraic manipulation, eventually simplifies the power series expression to match our original series' structure when \( x \) is chosen accordingly. Mastering differentiation within series provides insight into the behavior and summation of terms by leveraging derivative's power in unfolding series patterns.

Series Summation

Series summation is the process of adding up all terms in a sequence to find a total or closed form. In our exercise, after transforming the geometric series and differentiating steps, the ultimate goal was to substitute \( x = \frac{1}{2} \) to evaluate the series:

• By substituting \( x = \frac{1}{2} \), the expression simplifies to an easily calculable form, specifically leading us to determine the sum of \( \sum_{n=1}^{\infty} n^2 \left( \frac{1}{2} \right)^n \).
• This is done by plugging the values into the derived equation: \( \frac{1 + x}{(1-x)^3} \) results in \( \frac{1}{\left( \frac{1}{2} \right)^3} = 8 \).

Series summation involves clever manipulation of algebraic operations to find sums that aren't immediately obvious. Understanding each step ensures students can handle complex series assignments with ease, turning infinite seemingly puzzling problems into solvable solutions.