Problem 41
Question
$$ \text { Prove: If } \sum_{k=1}^{\infty} a_{k} \text { diverges, so does } \sum_{k=1}^{\infty} c a_{k} \text { for } c \neq 0 \text { . } $$
Step-by-Step Solution
Verified Answer
If \( \sum_{k=1}^{\infty} a_k \) diverges, \( \sum_{k=1}^{\infty} c a_k \) diverges for \( c \neq 0 \).
1Step 1: Identify the premises
We are given that the series \( \sum_{k=1}^{\infty} a_k \) diverges. We need to prove that \( \sum_{k=1}^{\infty} c a_k \) also diverges for any non-zero constant \( c \). This means we assume that the original series does not converge to a finite value.
2Step 2: Constant multiplication property
When a series \( \sum_{k=1}^{\infty} a_k \) diverges, any non-zero constant \( c \) multiplied by every term of this series, \( \sum_{k=1}^{\infty} c a_k \), retains the divergence unless the constant is zero, which is ruled out by our assumptions.
3Step 3: Contradiction method
Assume, for the sake of contradiction, that \( \sum_{k=1}^{\infty} c a_k \) converges. Then, by linearity of the series, if \( c eq 0 \), \( \sum_{k=1}^{\infty} a_k = \frac{1}{c} \sum_{k=1}^{\infty} c a_k \) must also converge. This contradicts our given condition that \( \sum_{k=1}^{\infty} a_k \) diverges.
4Step 4: Conclusion
Since assuming the convergence of \( \sum_{k=1}^{\infty} c a_k \) leads to a contradiction, our assumption must be false. Therefore, \( \sum_{k=1}^{\infty} c a_k \) must diverge.
Key Concepts
Constant Multiplication PropertyLinearity of SeriesContradiction Method
Constant Multiplication Property
The Constant Multiplication Property is a useful concept when dealing with series and their behavior. When you have a series \( \sum_{k=1}^{\infty} a_k \) that diverges, it essentially means that the sum doesn't settle at a finite number. The property states that if you take this divergent series and multiply each term by a constant \( c \), which isn't zero, the resulting series \( \sum_{k=1}^{\infty} c a_k \) will also diverge.
This happens because multiplying by a constant does not change the essential behavior of the series. If \( \sum_{k=1}^{\infty} a_k \) is off running to infinity or behaves erratically, doing the same thing while adjusting the scale (with a constant multiplier) won't suddenly cause it to converge. The key point is that the constant shifts every term equally but doesn't impose any convergent behavior.
This happens because multiplying by a constant does not change the essential behavior of the series. If \( \sum_{k=1}^{\infty} a_k \) is off running to infinity or behaves erratically, doing the same thing while adjusting the scale (with a constant multiplier) won't suddenly cause it to converge. The key point is that the constant shifts every term equally but doesn't impose any convergent behavior.
- Important note: This rule doesn't apply if the constant is zero, as multiplying by zero would make every term zero, leading to a series that trivially converges to zero.
Linearity of Series
Linearity of Series is a powerful tool that simplifies the manipulation of series. This property states two main points: you can distribute multiplication over addition and you can separate sums into two or more distinct pieces.
For instance, given a series \( \sum_{k=1}^{\infty} a_k \) that is manipulated by a constant \( c \), linearity allows us to express \( \sum_{k=1}^{\infty} c a_k \) by pulling out the constant as \( c\sum_{k=1}^{\infty} a_k \).
Now, this is critical: by the linearity of series, if we assume \( \sum_{k=1}^{\infty} c a_k \) converges and \( c eq 0 \), it forces \( \sum_{k=1}^{\infty} a_k \) to adopt the convergent behavior. However, if \( \sum_{k=1}^{\infty} a_k \) diverges, a contradiction arises, showing the initial assumption was incorrect.
For instance, given a series \( \sum_{k=1}^{\infty} a_k \) that is manipulated by a constant \( c \), linearity allows us to express \( \sum_{k=1}^{\infty} c a_k \) by pulling out the constant as \( c\sum_{k=1}^{\infty} a_k \).
Now, this is critical: by the linearity of series, if we assume \( \sum_{k=1}^{\infty} c a_k \) converges and \( c eq 0 \), it forces \( \sum_{k=1}^{\infty} a_k \) to adopt the convergent behavior. However, if \( \sum_{k=1}^{\infty} a_k \) diverges, a contradiction arises, showing the initial assumption was incorrect.
- This underlines the principle that linear operations, like multiplying by a constant, preserve the general characteristics of the series.
Contradiction Method
The Contradiction Method is a logical approach often used in proofs, including those involving series. It begins by assuming the opposite of what you want to prove. If this assumption logically leads to a contradiction—a statement known to be false—then your initial assumption must be wrong, thus proving your original claim.
In this series context, we start with the assumption that \( \sum_{k=1}^{\infty} c a_k \) converges, while we know that \( \sum_{k=1}^{\infty} a_k \) diverges. By applying linearity, if \( \sum_{k=1}^{\infty} c a_k \) converges, then \( \sum_{k=1}^{\infty} a_k \) should too, since \( c eq 0 \). But this contradicts our given information that the original series diverges.
In this series context, we start with the assumption that \( \sum_{k=1}^{\infty} c a_k \) converges, while we know that \( \sum_{k=1}^{\infty} a_k \) diverges. By applying linearity, if \( \sum_{k=1}^{\infty} c a_k \) converges, then \( \sum_{k=1}^{\infty} a_k \) should too, since \( c eq 0 \). But this contradicts our given information that the original series diverges.
- Thus, the contradiction shows that the assumption of convergence was flawed, cementing that \( \sum_{k=1}^{\infty} c a_k \) must indeed diverge.
Other exercises in this chapter
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