Problem 2 - Calculus

Question

Compute \(\sum_{j=0}^{3}\left(j^{2}-j\right)\) and \(\sum_{j=1}^{n} 1 / 2^{j}\)

Step-by-Step Solution

Answer

The first sum is 8, and the second sum is \(1 - \frac{1}{2^n}\).

Step 1: Identify the First Series

We need to compute the sum \(\sum_{j=0}^{3}(j^2 - j)\). This means we will calculate each term from \(j=0\) to \(j=3\), then add them together.

Step 2: Calculate Each Term of the First Series

Calculate each term individually: \(j=0: 0^2 - 0 = 0\), \(j=1: 1^2 - 1 = 0\), \(j=2: 2^2 - 2 = 2\), \(j=3: 3^2 - 3 = 6\).

Step 3: Sum the Terms of the First Series

Now add the terms: \(0 + 0 + 2 + 6 = 8\). So, \(\sum_{j=0}^{3}(j^2 - j) = 8\).

Step 4: Identify and Analyze the Second Series

We need to compute the sum \(\sum_{j=1}^{n} \frac{1}{2^j}\). It's a geometric series with the first term \(a = \frac{1}{2}\) and common ratio \(r = \frac{1}{2}\).

Step 5: Use the Formula for Geometric Series

The sum of the first \(n\) terms of a geometric series is \( S = \frac{a(1 - r^n)}{1 - r} \). Substitute \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\): \[ S = \frac{\frac{1}{2}(1 - (\frac{1}{2})^n)}{1 - \frac{1}{2}} = 1 - \frac{1}{2^n} \]

Step 6: Conclusion for the Second Series

Therefore, \(\sum_{j=1}^{n} \frac{1}{2^j} = 1 - \frac{1}{2^n}\). This gives the sum of the series for any \(n\).

Key Concepts

Series Summation

Series summation is the process of adding up all the terms in a series. A series is a sequence of numbers where each term is connected by a specific rule. In the exercise, we tackled two series. The first series involved summing terms calculated using the formula \(j^2 - j\), from \(j = 0\) to \(j = 3\). Here’s how it works:

• Find each term using the given formula.
• For example, when \(j = 0\), the term is \(0^2 - 0 = 0\); when \(j = 1\), it’s \(1^2 - 1 = 0\).
• Add all these calculated terms together (like \(0 + 0 + 2 + 6\)) to reach the total sum, which in this case is 8.

Series summation is useful in many areas of mathematics and science when you need to calculate total amounts.
Understanding this process helps in grasping complex mathematical concepts systematically.

Geometric Progression

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the exercise, the second series is a geometric progression where each term is of the form \(\frac{1}{2^j}\). Let's break it down:

• The first term \(a\) is \(\frac{1}{2}\).
• The common ratio \(r\) is also \(\frac{1}{2}\).
• Each subsequent term is simply half of the previous term.

Geometric progressions can either converge or diverge depending on the value of the common ratio. If the ratio is between -1 and 1, the series converges, often towards a finite sum, as seen in this exercise.
They are essential in understanding patterns in growth and decay models, like population growth or radioactive decay.

Mathematical Series

A mathematical series is the sum of the terms of a sequence. It can be finite or infinite. The two series from the exercise are examples of finite series as they have a definite number of terms to add. Here’s a general approach to tackling mathematical series:

• Identify the pattern or rule that defines the sequence.
• Calculate each individual term if necessary.
• Sum up all the terms to find the total, or use a formula for ease if available.

Mathematical series are pivotal in calculus and other branches of mathematics for approximating complex functions and calculating precise areas under curves.
Studying them enhances solving skills for real-world problems related to summation.

Sums of Sequences

Sums of sequences involve the addition of all the numbers in a series derived from a sequence. In our exercise, after applying the rules for calculating each term, the next step was to add them up. Each sequence can be infinite with a rule that continues endlessly, or finite with a clear endpoint. Some important points:

• Recognize whether the sequence is arithmetic or geometric, as this changes the approach.
• For geometric sequences, use the formula for the sum, which is often necessary when dealing with infinite sequences to manage a finite sum.
• In the case of \( \sum_{j=1}^{n} \frac{1}{2^j} \), understanding the geometric properties allows quick summation.

By mastering the sums of sequences, calculations involving series become straightforward and methodical.
This knowledge is directly applicable, not just in mathematics, but in fields like computer science, finance, and physics.