Problem 52

Question

If $\Sigma a_{n}$ is divergent and $c \neq 0,$ show that $\Sigma c a_{n}$ is divergent.

Step-by-Step Solution

Verified

Answer

$\Sigma c a_n$ is divergent because multiplying by a non-zero constant doesn't change divergence.

1Step 1: 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.

2Step 2: 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.

3Step 3: 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.

4Step 4: 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 $\Sigma c a_n$.

Key Concepts

Partial SumsScalar MultiplicationProperties of Series

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.

Understanding the partial sums is crucial, as they form the backbone of determining the ultimate nature of the series. In this context, since the series $ \Sigma a_n $ is divergent, its partial sums do not converge, indicating the series continually grows or oscillates indefinitely.

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’t 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.

Understanding scalar multiplication helps emphasize that scaling impacts size but not the convergent behavior of a series.

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 $.

These properties showcase that while operations like scalar multiplication can alter the magnitude of terms and partial sums, they do not change an underlying divergent nature to convergent. Understanding these helps ensure accurate predictions and interpretations in series analysis.

Problem 51

Problem 52

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