Question 4.T.1: (i) If the series ∑xn and ∑yn are both convergent, then the ...

(i) If the series \sum{x_{n}} and \sum{y_{n}} are both convergent, then the series \sum{(x_{n} + y_{n})}  is also convergent, and

\sum\limits_{n=1}^{∞}{(x_{n} + y_{n})} = \sum\limits_{n=1}^{∞}{x_{n}} + \sum\limits_{n=1}^{∞}{y_{n}}.

(ii) If the series \sum{x_{n}} is convergent and c is any real number, then \sum{cx_{n}} is convergent, and