[The Algebraic Limit Theorem for Series (2.7.1)] If \(\sum_{k=1}^{\infty} a_k = A\) and \(\sum_{k=1}^{\infty} b_k = B\), then
  1. \(\sum_{k=1}^{\infty} ca_k = cA\) and
  2. \(\sum_{k=1}^{\infty} (a_k + b_k) = A + B\).


For the absolute value function definition and other properties see here.
For the definitions of sequences and what it means to for a sequence to converge, see this, for subsequences see this.
For the definitions of series, and what it means to for a series to converge, see this.
For the “show the limit” template and an example, see this.

Formal Proof

To show that \(\sum_{k=1}^{\infty} ca_k = cA\), this means that we need to prove that the sequence of partial sums converges to \(cA\). In other words, \(\lim cs_m = cA\) where \((s_m)\) is the sequence of partial sums and \(s_m\) is defined as,

$$ \begin{align*} t_m = ca_1 + ca_2 + ca_3 + .... + ca_m, \end{align*} $$

But we are given that \(\sum_{k=1}^{\infty} a_k = A\). This means that the sequence of partial sums \((s_m)\) where \(s_m\) is defined as

$$ \begin{align*} s_m = a_1 + a_2 + a_3 + .... + a_m, \end{align*} $$

converges to \(A\). Since \(\lim s_m = A\), then by Algebraic Limit Theorem for Sequences, we have \(\lim c s_m = c A\) as required.

To prove (ii), we are given that \(\sum_{k=1}^{\infty} a_k = A\) which means that the sequence of partial sums converges to \(A\) or \(\lim s_m = A\). Similarly, \(\lim t_m = B\) where \((t_m)\) is the sequence of partial sums for the series \(\sum_{k=1}^{\infty} b_k\). Since \(\lim s_m = A\) and \(\lim t_m = B\), then by the Algebraic Limit Theorem, we can conclude that \(\lim s_m + t_m = A + B\). This implies that the series \(\sum_{k=1}^{\infty} (a_k + b_k) = A + B\) by definition. \(\blacksquare\)

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