Problem 111 - Essential Calculus

Question

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. For the alternating series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n},\) the partial sum \(S_{100}\) is an overestimate of the sum of the series.

Step-by-Step Solution

Answer

The statement is False. According to the Alternating Series Test, since our series commences with a negative term, the partial sum at any even \(n\) is an underestimate of the total sum. Thus, \(S_{100}\) being an even numbered partial sum is an underestimate, not an overestimate.

Step 1: Define The Series

The given series is an alternating series defined as \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\). It is important to notice that it's an alternating series because it contains the term \((-1)^n\), which means the signs of the terms alternate between positive and negative.

Step 2: Understand The Statement

The statement to evaluate is that the partial sum \(S_{100}\) is an overestimate of the total sum of the series. This statement is saying that if we were to add up the first 100 terms of the series, this sum would be greater than the total sum of the infinite series.

Step 3: Evaluate The Statement

The Alternating Series Test states that if an alternating series satisfies two conditions - the terms are decreasing and the limit as \(n\) approaches infinity is zero - the successive partial sums of the series will alternate between being overestimates and underestimates of the actual total sum. Here, our series does indeed satisfy these conditions. However, since our first term is negative when \(n=1\), the partial sum at any even \(n\) is an underestimate of the total, because the series dropped below the total on the last negative term. On the contrary, the partial sum at any odd \(n\) is an overestimate, since the series rose above the total on the last positive term. Following this, as \(100\) is an even number, the partial sum \(S_{100}\) is an underestimate, not an overestimate, for the series.

Key Concepts

Partial Sums

When we talk about partial sums in mathematics, we're dealing with the sum of the first few terms of a series. For instance, if you have a series, you only add up a specific number of terms, say the first 100. This is called the 100th partial sum, denoted as \(S_{100}\). Partial sums provide a way to approximate the sum of an entire series by just adding a finite number of terms.

In an alternating series like \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\), the partial sum can be tricky. The "alternating" nature means the terms switch from positive to negative and back again. This alternation affects whether the partial sum tends to overshoot or undershoot the true infinite sum. Understanding partial sums allows us to see how a series behaves before adding infinitely many terms, which is crucial for making approximations or understanding convergence.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. You've probably seen it written like \(\sum_{n=1}^{\infty} a_n\), which means you keep adding the terms \(a_n\) forever. The big question with infinite series is whether they converge, which means they approach a limit.

In the context of the given series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\), convergence is essential. This series is a prime example of an alternating series, and it's important to know that not all infinite series converge. The convergence of alternating series often hinges on their terms continually getting smaller and smaller, approaching zero as we add more terms, a condition needed for this specific series to converge.

Infinite series might begin with random-looking numbers, but as you keep adding terms, they start settling into a pattern. Sometimes they stabilize at a definite number, signaling convergence. Other times, they wander without settling, indicating divergence, which means they don't sum to a finite value.

Alternating Series Test

The Alternating Series Test provides a helpful criterion to determine whether an alternating series converges. To apply this test, the series should meet two key conditions:

• The terms \(b_n\), ignoring signs, must be decreasing, i.e., \(b_{n+1} \leq b_n\).
• The limit of \(b_n\) as \(n\rightarrow\infty\) must be zero.

The series \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\) meets both of these conditions. The terms \(\frac{1}{n}\) decrease as \(n\) gets bigger, and their limit is zero as they approach infinity.

Because of these properties, the test confirms convergence. The behavior of partial sums maneuvering around the true sum further underscores this phenomenon. The test reveals that these partial sums will alternate around the correct total sum, being either slightly above or below it, depending on whether they have an even or odd number of terms. This alternating behavior is what makes predicting the partial sums' relationship to the true sum so interesting and sometimes challenging.