Question

Prove that if \(\sum a_{k}\) diverges, then \(\sum c a_{k}\) also diverges, where \(c \neq 0\) is a constant.

Short answer

Question: Prove that if the series \(\sum a_{k}\) diverges and \(c\) is a nonzero constant, then the series \(\sum c a_{k}\) also diverges. Answer: We proved that given the divergent series \(\sum a_{k}\) and a nonzero constant \(c\), the series \(\sum c a_{k}\) has partial sums with a limit that does not exist or is not finite as \(n\) approaches infinity. Thus, by definition, the series \(\sum c a_{k}\) also diverges when \(c \neq 0\).

Step-by-step solution

Recall the definition of a divergent series

A series \(\sum a_{k}\) is said to diverge if the partial sums \(S_n = a_1 + a_2 + \cdots + a_n\) do not converge as \(n\) approaches infinity. In other words, the series diverges if \(\lim_{n \to \infty} S_n\) does not exist or is not finite.

Calculate the partial sums of the series \(\sum c a_{k}\)

Let \(T_n\) denote the partial sums of the series \(\sum c a_{k}\). That is, \(T_n = c a_1 + c a_2 + \cdots + c a_n = c (a_1 + a_2 + \cdots + a_n) = c S_n\).

Consider the limit of \(T_n\) as \(n\) approaches infinity

Since the series \(\sum a_{k}\) diverges by assumption, then \(\lim_{n \to \infty} S_n\) does not exist or is not finite. We need to show that the limit of the sums of the series \(\sum c a_{k}\), which is \(\lim_{n \to \infty} T_n = \lim_{n \to \infty} c S_n\), also does not exist or is not finite. Since \(c\) is a nonzero constant, we can factor out \(c\) from the limit as follows: \(\lim_{n \to \infty} c S_n = c \lim_{n \to \infty} S_n\).

Show that the limit of partial sums of \(\sum c a_{k}\) does not exist or is not finite

We know that \(\lim_{n \to \infty} S_n\) does not exist or is not finite since the series \(\sum a_{k}\) diverges. Then, the product of a nonzero constant \(c\) with a limit that does not exist or is not finite also does not exist or is not finite. Therefore, we have shown that \(\lim_{n \to \infty} c S_n\) does not exist or is not finite.

Conclude that \(\sum c a_{k}\) is a divergent series

We have shown that the partial sums of the series \(\sum c a_{k}\), which are given by \(T_n = c S_n\), have a limit that does not exist or is not finite as \(n\) approaches infinity: \(\lim_{n \to \infty} T_n = \lim_{n \to \infty} c S_n\) does not exist or is not finite. Thus, by definition, the series \(\sum c a_{k}\) also diverges when \(c \neq 0\).

Key concepts

Divergent Series

A divergent series is a sequence of numbers whose partial sums do not settle on a single value as you keep adding more terms. In simpler terms, if you keep adding up the numbers in the series, and you notice that your total either keeps growing endlessly or does not approach a steady number, then the series is divergent. This is in contrast to a convergent series, where the sum approaches a specific number as more terms are added.
Understanding divergent series helps build the foundation for recognizing when calculations might "go off to infinity" or "bounce around" without arriving at a particular value. Divergent series are important in calculus because they highlight limits of certain processes and provide insight into the behavior of infinity.
Generally, in math, if a series is said to be divergent, it simply means that as the number of terms grows infinitely, the sum does not stabilize, indicating that the series doesn’t converge.

Partial Sums

Partial sums are a central concept when dealing with series. The partial sum, symbolized often as \( S_n \), is what you get when you sum only a finite number of terms from a series. For instance, if you start with the first term, add the second, then the third, and keep going until a specific point \( n \), you've calculated a partial sum.
When determining whether a series converges or diverges, we look at what happens to these partial sums as \( n \) becomes very large. In mathematical language, we examine the behavior of \( S_n \) as \( n \rightarrow \infty \). If these sums level out to some number, the series converges; if they don't, such as in our starting problem of transverse series, this signals divergence.
In our problem example, \( S_n \) eventually becomes \( T_n = cS_n \) for the new series \( \sum ca_k \); recognizing this helps us directly apply conclusions about divergence from the original series.

Infinity Limit

The concept of limits extending to infinity is crucial in calculus when exploring the bounds of sequences and series. When we say a limit reaches infinity, symbolically \( \lim_{n \to \infty} S_n \), it's akin to analyzing whether a sequence grows indefinitely without ever settling down.
Limits enable us to mathematically express and deal with behaviors that don't fit neatly within finite bounds. If a sequence or its derivatives point toward infinity, it often signals that the underlying process or function behaves unboundedly in that direction, again, indicating divergence.
In connection to the partial sums, if \( S_n \to \infty \) or simply does not approach any real number as \( n \to \infty \), we declare that the series is divergent, substantiating why the multiplication by a constant doesn’t change its divergence nature.

Constant Multiplication Effect

The multiplication of a series by a constant, \( c eq 0 \), significantly impacts its behavior without changing its fundamental nature if the series is divergent. In our exercise, this principle demonstrates that multiplying each term by a constant essentially scales the entire series but doesn’t alter convergence or divergence outcomes.
For instance, if the original series, \( \sum a_k \), diverges, multiplying by \( c \) leads to \( T_n = cS_n \). When checking its limits, we find that scaling doesn't transform a divergent outcome to convergent or vice versa. Why?

• If the original sum doesn't settle or heads toward infinity, amplifying it by any constant keeps that trend intact.
• An infinite or non-existing limit, when multiplied by a non-zero constant, remains infinite or non-existent, maintaining divergence.

Understanding this constant multiplication effect simplifies solving complex problems where the properties of convergence and divergence play critical roles.