Problem 8 - Calculus: One and Several Variables

Question

Find the sum of the series. $$\sum_{k=0}^{\infty} \frac{1}{2^{k+3}}$$

Step-by-Step Solution

Answer

The sum of the convergent geometric series \(\sum_{k=0}^{\infty} \frac{1}{2^{k+3}}\) is \(S = \frac{1}{4}\).

Step 1: Identifying the Terms of the Geometric Series

The given series is: $$ \sum_{k=0}^{\infty} \frac{1}{2^{k+3}} $$ We can rewrite this series as: $$ \sum_{k=0}^{\infty} \left(\frac{1}{2^3}\right)\left(\frac{1}{2}\right)^k $$ Now, we can clearly see that the first term `a` is \(\frac{1}{2^3}\) and the common ratio `r` is \(\frac{1}{2}\).

Step 2: Applying the Convergent Geometric Series Formula

Now that we have identified `a` and `r`, we can use the formula for the sum of a convergent geometric series: $$ S = \frac{a}{1 - r} $$ Substitute the values `a = \frac{1}{2^3}` and `r = \frac{1}{2}` into the formula: $$ S = \frac{\frac{1}{2^3}}{1 - \frac{1}{2}} $$

Step 3: Simplifying the Expression

Next, we will simplify the expression: $$ S = \frac{\frac{1}{8}}{\frac{1}{2}} $$ To divide the fractions, we can multiply by the reciprocal: $$ S = \frac{1}{8} \cdot \frac{2}{1} $$ This simplifies to: $$ S = \frac{1}{4} $$ The sum of the given series converges to \(\frac{1}{4}\).

Key Concepts

Convergent Series

When we talk about convergent series, we refer to a series whose terms approach a specific value as they progress indefinitely. In simpler terms, it's like a sequence of numbers where the total sum gets closer and closer to a certain number, instead of going off to infinity. For example, in our exercise, the series \( \sum_{k=0}^{\infty} \frac{1}{2^{k+3}} \) is convergent.
Here’s why:

• In a convergent series, the common ratio (r) is crucial. For a geometric series to converge, the absolute value of \( r \) must be less than 1, \( |r| < 1 \).
• This ensures that each subsequent term in the series becomes smaller, allowing the sum to stabilize towards a particular value.

In our case, \( r = \frac{1}{2} \) fits this rule because \( \left|\frac{1}{2}\right| = 0.5 < 1 \), indicating that this series converges. Hence, by employing the formula for a convergent geometric series \( S = \frac{a}{1 - r} \), we get closer and closer to the sum of \( \frac{1}{4} \).
Understanding convergent series is key when dealing with infinite sequences, as it helps determine whether the series has a finite sum or not.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Think of it as an endless addition problem where you add up numbers without ever stopping. They are written using the summation notation \( \sum \), which is a compact way to represent all the terms in the sequence.
In our problem, the series is written as \( \sum_{k=0}^{\infty} \frac{1}{2^{k+3}} \). This notation means that we're adding up terms from \( k=0 \) to \( k=\infty \), or infinitely many terms:

• As \( k \) increases, each term in this series becomes smaller and smaller, thanks to the structure of the series.
• The initial terms affect the overall sum significantly more than those that come later because they contribute more to the total before the terms become vanishingly small.

Infinite series can be either convergent or divergent. Convergent series, like ours, settle towards a finite value. This is different from a divergent series, where the sum keeps growing larger without bound. The fascinating part about some infinite series is that they have a finite sum, despite having infinitely many terms.

Geometric Progression

A geometric progression is a sequence where each term after the first is derived by multiplying the previous one by a fixed, non-zero number known as the common ratio. In math, it looks like this: if the first term is \( a \), then the sequence progresses as \( a, ar, ar^2, ar^3, \ldots \).
In our exercise, the series \( \sum_{k=0}^{\infty} \frac{1}{2^{k+3}} \) exemplifies a geometric progression because:

• The first term \( a \) is \( \frac{1}{2^3} \).
• The common ratio \( r \) is \( \frac{1}{2} \), meaning each term is half the size of the term before it.

Understanding geometric progressions helps us identify and analyze series more efficiently. Because they have such a regular pattern, they are easier to sum up, especially when dealing with infinite terms. The general formula for the sum of an infinite geometric series is \( S = \frac{a}{1-r} \). This relies on \( |r| < 1 \) for convergence, ensuring the series stays within finite bounds. This simple, predictable pattern makes geometric progressions a powerful tool in mathematics.