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Magazine Chapter 8: Problem 52
If \(\Sigma a_{n}\) is divergent and \(c \neq 0,\) show that \(\Sigma c a_{n}\) is divergent.
Short Answer
Expert verified
\(\Sigma c a_n\) is divergent because multiplying by a non-zero constant doesn't change divergence.
Step by step solution
01
Understand the Definitions
In this problem, a series \(\Sigma a_n\) is referred to as divergent. A series is divergent if the sequence of its partial sums does not converge to a finite limit. This means that as you add more terms \(a_n\) together, the sum grows indefinitely or does not settle down to a particular value.
02
Scalar Multiplication and Series
We are given a constant \(ceq 0\). We need to understand that when each term \(a_n\) of the series is multiplied by a non-zero constant \(c\), the series becomes \(\Sigma c a_n\). The nature of the series \(\Sigma a_n\) being divergent will be preserved in \(\Sigma c a_n\) because scalar multiplication by a constant factor affects the size of the terms but not their tendency to converge or diverge.
03
Apply Properties of Series
A fundamental property of series says that if \(\Sigma a_n\) is divergent and \(c eq 0\), then multiplying each term of the series by \(c\) leads to \(\Sigma c a_n\) also being divergent. Because the multiplication will proportionally scale each partial sum by \(c\), this increases (or decreases if \(c<0\)) the magnitude of the partial sums, but does not change the fact that they do not settle to a finite value.
04
Conclude the Divergence of the Series
Since the original series \(\Sigma a_n\) is divergent and multiplying by any non-zero scalar \(ceq 0\) preserves this divergence, the series \(\Sigma c a_n\) must also be divergent. Thus, if the sequence of partial sums of \(\Sigma a_n\) diverges, so does the sequence of partial sums of \(\Sigma c a_n\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Partial Sums
In mathematics, when dealing with series, a vital concept to understand is *partial sums*. A partial sum is the sum of the first "n" terms of a sequence. For a given series \( \Sigma a_n \), the partial sum \( S_n \) is given by \( S_n = a_1 + a_2 + \ldots + a_n \). This sequence of partial sums \( \{S_n\} \) provides insight into the behavior of the series as a whole.
- Convergence: If the sequence of partial sums settles to a specific number as "n" becomes very large, we say the series is convergent.
- Divergence: Conversely, if the partial sums grow without bound or oscillate, failing to settle on a finite number, the series is categorized as divergent.
Scalar Multiplication
Scalar multiplication in the context of series involves multiplying each term of the series by a constant. Consider a series \( \Sigma a_n \). If we introduce a non-zero constant \( c \), the transformed series becomes \( \Sigma c a_n \). This operation doesn鈥檛 alter the core characteristic of being divergent or convergent, but rather scales the terms and their sum.
- Preserved Divergence: If \( \Sigma a_n \) is divergent, multiplying by \( c eq 0 \) means \( c \times (a_1 + a_2 + \cdots + a_n) = cS_n \), which simply scales the partial sums by \( c \). Thus, the divergence remains, as the behavior of summing indefinitely or failing to settle persists regardless of the scaling factor.
- Significance of \( c \): The constant \( c \) can affect the magnitude of sums significantly but does not turn a divergent series into a convergent one or vice versa.
Properties of Series
Properties of series provide a framework to understand and predict the behavior of series operations. When handling the divergence and convergence of series, these properties are invaluable:
- Scale Invariance of Divergence: As seen, if \( \Sigma a_n \) is divergent, \( \Sigma c a_n \) remains divergent for any non-zero constant \( c \). This property confirms that the behavior of not settling to a finite value endures, irrespective of scaling.
- Linear Combination: If the series involves linear combinations, such as adding two divergent series, the resulting series is typically divergent. In contrast, the convergence depends on more strict conditions than multiplication by a constant.
- Effect on Partial Sums: The divergent nature of \( \Sigma a_n \), ensured by the divergent partial sums, equally applies to \( \Sigma c a_n \), where the partial sums are merely the original sums scaled by \( c \).
