Question

A series is given. (a) Find a formula for \(S_{n},\) the \(n^{\text {th }}\) partial sum of the series. (b) Determine whether the series converges or diverges. If it converges, state what it converges to. $$\sum_{n=1}^{\infty} 2$$

Short answer

The series diverges since its partial sum grows indefinitely.

Step-by-step solution

Understand the Series

The given series is \( \sum_{n=1}^{\infty} 2 \). This means every term of the series is the constant value 2. Hence, this is an infinite series of repeating 2's.

Define the Partial Sum Formula

To find \( S_{n} \), the \( n^{\text{th}} \) partial sum of the series, we sum the first \( n \) terms of the series. Since each term is 2, the \( n \) terms sum to \( 2 + 2 + \cdots + 2 = 2n \). Therefore, \( S_{n} = 2n \).

Determine Series Convergence or Divergence

A series converges if its sequence of partial sums \( S_{n} \) approaches a finite limit as \( n \to \infty \). Here, \( S_{n} = 2n \). As \( n \to \infty \), \( S_{n} \to \infty \). Thus, the series does not approach a finite limit.

State Conclusion

Since the partial sum \( S_{n} \) grows without bound, the series \( \sum_{n=1}^{\infty} 2 \) diverges. Therefore, it does not converge to any finite sum.

Key concepts

Partial Sum

Partial sums are like landmarks in the world of series. They help us understand the series step by step. The partial sum \( S_n \) is the sum of the first \( n \) terms in a series. For example, in our given series \( \sum_{n=1}^{\infty} 2 \), each term is the number 2. The formula for the \( n^{\text{th}} \) partial sum of this series is simply \( S_n = 2 + 2 + \cdots + 2 = 2n \). We are just adding the same number over and over.

• \( n = 1 \): The first partial sum is \( 2 \times 1 = 2 \)
• \( n = 2 \): The second partial sum is \( 2 + 2 = 2 \times 2 = 4 \)
• \( n = 3 \): The third partial sum is \( 2 + 2 + 2 = 2 \times 3 = 6 \)

Notice the pattern? As \( n \) increases, we just multiply \( n \) by 2. This showcases how using partial sums is a handy trick to see what's happening in a series.

Infinite Series

An infinite series is when you keep adding numbers forever. There is no end, unlike a finite series. In our case, the series \( \sum_{n=1}^{\infty} 2 \) consists of adding the value 2, repeatedly, without stopping.Infinite series can be overwhelming, but they offer a unique way to explore how numbers behave. Here are some characteristics:

• Each term you add stays the same. It's 2 in each position of the series.
• Think of it like a running meter, continuously adding two without a stop.
• Since it's "infinite," we don’t count "how many," but we study the impact of endlessly adding.

Infinity is big, so understanding infinite series helps us grasp mathematical concepts that seem too enormous for regular counting.

Convergence and Divergence

When we talk about convergence and divergence, we’re essentially asking: "What happens when we keep adding things forever?" Convergence refers to a series that eventually settles towards a specific number. On the other hand, divergence describes a series that never settles down, like our example: \( \sum_{n=1}^{\infty} 2 \).The key to determining if a series converges or diverges is looking at the partial sums:

• If the partial sum \( S_n \) gets closer and closer to a specific number, the series converges.
• If \( S_n \) keeps growing without settling down, the series diverges.

For our specific series, \( S_n = 2n \). As \( n \) gets larger, \( S_n \) keeps growing towards infinity. There's no finite number it approaches, so the series diverges. Understanding whether a series converges or diverges helps us know if there's a way to "catch" the value of an entire infinite process.