If $\sum_{n=1}^{\infty}|f_n|$ is integrable, then:
(a) Each $f_n$ is integrable
(b) $\sum_{n=1}^{\infty}f_n$ converges a.e. and is integrable
(c) $\int_\Bbb{R}\sum_{n=1}^{\infty}f_nd\mu = \sum_{n=1}^{\infty}\int_\Bbb{R}f_nd\mu$
Since $\sum_{n=1}^{\infty}|f_n|$ is integrable, we have that $\int|f_n|\leq \int\sum_{n=1}^{\infty}|f_n|<\infty$ and that each $\int|f_n|$ is finite hence, $f$ is integrable.
But I'm lost here. Any help is appreciated.
Since $\sum_{n=0}^{\infty} |f_n|$ is integrable then $\sum_{n=0}^{\infty} |f_n| < \infty$ a.e, that is, the series $\sum_{n=0}^{\infty} f_n$ converges absolutely a.e $\implies$ converges a.e. Now, if we put $g_n = \sum_{k=0}^{n} f_k$ a.k.a the $n$-th partial sum of the series, we know that $g_n \to g = \sum_{n=0}^{\infty} f_n$ a.e. Moreover, $$ |g_n| = \left|\sum_{k=0}^{n} f_k\right| \leq \sum_{k=0}^{n} |f_k| \leq \sum_{k=0}^{\infty} |f_k|, $$ which is integrable. Then, by the Domianted Convergence Theorem, $g$ is integrable and $$ \lim_{n\to\infty}\int_{\mathbb{R}}\sum_{k=0}^{n} f_k \, d\mu = \lim_{n\to\infty}\int_{\mathbb{R}} g_n \, d\mu = \int_{\mathbb{R}}g \,d\mu = \int_{\mathbb{R}} \sum_{k=0}^{\infty} f_k \,d\mu. $$