Problem 8 - Discrete Mathematics with Applications

Question

Verify Theorem 10.1 using Exercise 2

Step-by-Step Solution

Answer

For the infinite geometric series \(\sum_{n=0}^{\infty} x^n\), we apply the formula \(S = \frac{a_1}{1 - r}\), with \(a_1 = x^0 = 1\) and r = x. Substituting the values, we get \(S = \frac{1}{1 - x}\), which converges when |x| < 1. Thus, Theorem 10.1 is verified using Exercise 2, as the series sum is \(\frac{1}{1-x}\).

Step 1: Write down the given infinite geometric series

Write down the given infinite geometric series: \[\sum_{n=0}^{\infty} x^n\]

Step 2: Apply the formula for the sum of an infinite geometric series

To find the sum of an infinite geometric series, we use the formula: \[S = \frac{a_1}{1 - r}\] Where S is the sum of the series, \(a_1\) is the first term, and r is the common ratio between consecutive terms. In our case, \(a_1 = x^0 = 1\) and r = x.

Step 3: Substitute the values into the formula

Substitute the values of \(a_1\) and r into the formula: \[S = \frac{1}{1 - x}\]

Step 4: Check if the series converges and the theorem holds

Since |x| < 1, the series converges, and we can see that the sum of the given infinite geometric series is \(\frac{1}{1-x}\). Hence, Theorem 10.1 is verified using Exercise 2.

Key Concepts

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Imagine you keep adding numbers together, without ever stopping. It’s like stacking blocks, but there are always more blocks to add! For a series to be useful, we often need to find its sum, even if it seems endless at first glance. In mathematical equations, an infinite series is usually written using the summation notation, for example, \[\sum_{n=0}^{\infty} x^n\] This notation means we start at \(n=0\) and continue indefinitely, adding each term \(x^n\) to find the collective sum.

Convergence

Convergence is a crucial concept when dealing with infinite series. A series converges if, as we keep adding more terms, we approach a specific number, rather than wandering off to infinity. Think of convergence like aiming a hose at a plant; you want the water to remain focused rather than spread everywhere.

For a geometric series, which is a series with a constant ratio, convergence happens if and only if the absolute value of the common ratio \(r\) is less than 1, \(|r| < 1\). This ensures that the terms are getting smaller, and the series tends towards a particular value, rather than continuing to grow or oscillate endlessly.

Mathematical Proof

Mathematical proof is essentially a logical argument, showing that a particular statement or theorem is undeniably true. This process involves several steps, much like solving a mystery, each leading from known truths to the conclusion you are proving.

To prove Theorem 10.1 using Exercise 2, you would perform calculations step by step. You start by writing the infinite series:\[\sum_{n=0}^{\infty} x^n\]Applying the formula, \(S = \frac{a_1}{1 - r}\), you substitute \(a_1\) as 1 and \(r\) as \(x\). Verifying each step with logical consistency is key to mathematical proof.

Theorem Verification

Theorem verification is when we confirm a theorem's validity through a demonstration or computation. In the simplest terms, it’s like double-checking your calculations to make sure everything adds up. Verification gives confidence in the theorem's reliability.

By using Exercise 2, we verified Theorem 10.1, confirming that the sum of the geometric series \(\sum_{n=0}^{\infty} x^n\) indeed equates to \(\frac{1}{1-x}\), given \(|x| < 1\). This theorem helps us manage infinite geometric series and is fundamental for understanding broader mathematical concepts. Such verification processes solidify our understanding, aligning theoretical math with observable results.