Let $\langle f_n\rangle$ be a sequence of functions from the space $L^p(X,\mathcal M,\mu)$, $1\le p<\infty$, such that the series $$ \sum_{n=1}^\infty \|f_n\|_p \ \ \ (*) $$ converges. Prove that then the series $$ \sum_{n=1}^\infty f_n(x) $$ converges in $L^p(X,\mathcal M,\mu)$.
I need to show that the sequence of partial sums $\{g_k(x)\}=\{ \sum_{n=1}^k f_n(x) \}$ is a Cauchy sequence in $L^p(X,\mathcal M,\mu)$, i.e. I need to show that for any $\epsilon > 0 $, there exists $M >> 0$ such that $n \geq m \geq M$ implies that $$||g_m(x)-g_n(x)||_p < \epsilon \ \ \ \text{or} \ \ \ \big(\int_X |\sum_{j=n+1}^m f_j(x)|^p d\mu \big)^{1/p}<\epsilon $$ But I do not see how to get this using the convergence of (*). I already appreciate your hints/answers.
This is true in any complete normed space. You have $$ \left\|\sum_{n=k}^mf_n\right\|_p\leq\sum_{n=k}^m\|f_n\|_p. $$ This allows you to show that the sequence of partial sums is Cauchy, by using the convergence of $(*) $.