Let $\sum a _ { n }$ And $\sum b _ { n }$

question 227

Essay

Let $\sum a _ { n }$ and $\sum b _ { n }$ be two series. Determine whether each of the following statements is true or false. Justify your answer.
(a) If $\sum a _ { n }$ converges, then $a _ { n } \rightarrow 0$ .
(b) If $a _ { n } \rightarrow 0$ , then $\sum a _ { n }$ converges.
(c) If $\sum a _ { n }$ converges, and $\sum b _ { n }$ diverges, then $\sum \left( a _ { n } + b _ { n } \right)$ diverges.
(d) If $\sum a _ { n }$ diverges, and $\sum b _ { n }$ diverges, then $\sum \left( a _ { n } + b _ { n } \right)$ diverges.
(e) If $\sum a _ { n }$ converges, and $\lim _ { n \rightarrow \infty } b _ { n } = 0$ then $\sum \left( a _ { n } + b _ { n } \right)$ converges.

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