Answer to Question #184536 in Real Analysis for Leonard

Question

1. Use the definition of the limit  to show that the sequence (1 + (−1)^n) is divergent.

Answer 1

$\begin{array}{l} \sum_{n=0}^{\infty}\left(1+(-1)^{n}\right) \rightarrow \text { divergent } \\ \text { Rule: We have } \sum_{k=0}^{\infty} a_{k} \text { , } \\ \text { if } \lim _{k \rightarrow \infty} a_{k}=0 \text { , the suries diverges } \\ \lim _{n \rightarrow \infty}\left(1+(-1)^{n}\right)=\lim _{n \rightarrow \infty} 1+\lim _{n \rightarrow \infty}(-1)^{n}= \\ =1+1=2 \\ n=2 k, k \in N \\ \text { the series diverges } \end{array}$