Property of divergent series Prove Property 2 of Theorem 10.8: If ∑ aₖ diverges, then ∑ caₖ also diverges, for any real number c ≠ 0.

Verified step by step guidance

1. Recall the statement of Property 2 of Theorem 10.8: If the series \(\sum a_k\) diverges, then for any real number \(c \neq 0\), the series \(\sum c a_k\) also diverges.
2. Start by assuming that \(\sum a_k\) diverges. This means the sequence of partial sums \(S_n = \sum_{k=1}^n a_k\) does not converge to a finite limit.
3. Consider the series \(\sum c a_k\) and its sequence of partial sums \(T_n = \sum_{k=1}^n c a_k\). By properties of sums, this can be written as \(T_n = c \sum_{k=1}^n a_k = c S_n\).
4. Since \(c \neq 0\), the behavior of \(T_n\) is directly related to \(S_n\). If \(S_n\) does not converge, then multiplying by a nonzero constant \(c\) will not produce a convergent sequence \(T_n\).
5. Therefore, the sequence of partial sums \(T_n\) of \(\sum c a_k\) also diverges, proving that \(\sum c a_k\) diverges whenever \(\sum a_k\) diverges and \(c \neq 0\).

Key Concepts

Divergent Series

A series is divergent if its sequence of partial sums does not approach a finite limit. This means the sum grows without bound or oscillates indefinitely, so it does not converge to a specific value.

Scalar Multiplication of Series

Multiplying each term of a series by a nonzero constant c scales the partial sums by c. This operation preserves the nature of convergence or divergence because scaling a divergent sequence by a nonzero factor cannot produce convergence.

Properties of Limits and Series

The limit of a sequence multiplied by a constant equals the constant times the limit of the sequence, if the limit exists. For series, this means if the original series diverges, scaling by a nonzero constant also results in divergence, as the limit of partial sums does not become finite.