Algebraic Limit Theorem for Series

If $\sum a_{n} = A$ and $b = \sum b_{n} = B$ and $c \in R$ . then:

Proof

t_{m} = c a_{1} + c a_{2} + \dots + c a_{m}

converges to $c A$ . We are just given $\sum_{k = 1}^{\infty} a_{k}$ converges to $A$ . So the partial sums:

s_{m} = a_{1} + a_{2} + \dots + a_{m}

converge to $A$ . Because $t_{m} = c s_{m}$ then using the Limit Laws (Algebraic Limit Theorem) yields $(t_{m}) \to c A$