Question

Theorem 4 shows that a convergent series does not change its sum if every several consecutive terms are replaced by their sum. Show by an example that the reverse process (splitting each term into several terms) may affect convergence.

Short answer

Splitting terms alters the sum, contrary to merging consecutive terms.

Step-by-step solution

Understanding the Theorem 4

Theorem 4 states that if you have a convergent series and you replace several consecutive terms with their sum, the series will still converge to the same limit. This means the operation does not alter the convergence.

Selecting a Convergent Series

To illustrate the reverse process, begin with a simple convergent series, such as the geometric series: \( \sum_{n=0}^{\infty} \left( \frac{1}{2} \right)^n \), which converges to \( 2 \) since the sum of an infinite geometric series \( S = \frac{a}{1-r} \).

Splitting Terms into Parts

Now, let's split each term of the series \( \left( \frac{1}{2} \right)^n \) into two terms, for example, \( \left( \frac{1}{4} \right)^n + \left( \frac{1}{4} \right)^n \), such that \( \left( \frac{1}{2} \right)^n = \left( \frac{1}{4} + \frac{1}{4} \right)^n \).

Form New Series

The new series would now be \( \sum_{n=0}^{\infty} \left( \frac{1}{4} \right)^n + \sum_{n=0}^{\infty} \left( \frac{1}{4} \right)^n \).

Convergence Check

Each component series \( \sum_{n=0}^{\infty} \left( \frac{1}{4} \right)^n \) converges individually to \( \frac{1}{1 - \frac{1}{4}} = \frac{4}{3} \). Therefore, the newly formed series, which is their sum, leads to a total sum of \( \frac{4}{3} + \frac{4}{3} = \frac{8}{3} \).

Conclusion and Contrast

Thus, the process of splitting terms alters the original series sum from \( 2 \) to \( \frac{8}{3} \), showing that the reverse operation indeed affects the convergence and sum of the series.

Key concepts

Series Transformation

Series transformation involves modifying the terms in a given sequence or series without changing its convergence properties. In convergent series transformation, it's essential to understand that combining consecutive terms into a single sum doesn't affect the overall sum or convergence of the series. This concept is supported by Theorem 4, which states that if you replace consecutive terms of a convergent series with their sum, the series will still converge to the same limit.

This property implies stability in the convergence of the series when such transformations are applied. However, the reverse process, where you split one term into multiple terms, does not guarantee that the series will exhibit the same convergence behavior or remain convergent. The reason is that series convergence depends heavily on the sum of the terms - changes in the sum of terms affect the behavior and limit of the series.

Thus, series transformation is a powerful tool for simplifying series, but must be applied with an understanding of its implications on the series' convergence.

Geometric Series

A geometric series is a series of the form \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio. The convergence of a geometric series is determined by the absolute value of the common ratio \( r \).

• If \( |r| < 1 \), the series converges to \( \frac{a}{1-r} \).
• If \( |r| \geq 1 \), the series diverges.

The geometric series is particularly interesting when it comes to series transformations. For instance, the series \( \sum_{n=0}^{\infty} \left( \frac{1}{2} \right)^n \) converges to \( 2 \) because the first term \( a = 1 \) and the common ratio \( r = \frac{1}{2} \) falls within the convergence condition (\( |r| < 1 \)).

By understanding the geometric formula used to compute the sum, it's possible to see how changing each term — even in a seemingly harmless way — can significantly alter the behavior of the entire series, emphasizing the necessity to navigate series transformations with caution.

Convergence and Divergence

Convergence and divergence are foundational concepts in calculus and series analysis. A series is said to converge if the sum of its terms approaches a finite number as more terms are aggregated. Conversely, a series diverges if the sum grows without bounds or oscillates indefinitely.

• Convergent Series: These have a finite limit as the number of terms approaches infinity.
• Divergent Series: These do not settle at a finite number. They either increase, decrease perpetually or swing between values.

A convergent series cannot be casually transformed without affecting its convergent properties. For example, by splitting terms in a geometric series, each term can effectively change its contribution to the total sum, thus potentially converting a convergent series into a divergent one or altering its existing sum, as shown in the exercise.

Understanding convergence and divergence allows one to determine the behavior of a series post transformation. It assists in predicting whether changes to the structure of a series — such as splitting terms or other modifications — will result in a series that still retains its convergent nature or moves it toward divergence.