Let $s=\sigma+it$ Let $s_0$ be such that $f \left({s_0}\right)$ converges.  Let $S\left(m,n\right) = \displaystyle \sum_{k \mathop = n}^m a_k e^{-\lambda_ks_0}$ We may create a new Dirichlet series that converges at 0 by writing: {{begin-eqn}} {{eqn | l = g \left({s}\right)       | r = f \left({s + s_0}\right)   }} {{eqn | r = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n \left({s + s_0}\right)} }} {{eqn | r = \sum_{n \mathop = 1}^\infty a_n e^{-\lambda_n s_0} e^{-\lambda_n s} }} {{end-eqn}} Thus it suffices to show $g \left({s}\right)$ converges for $\sigma>0$ By Cauchy's Convergence Criterion, it suffices to show that for all $\epsilon>0$ there exists an $N$ such that for all $m,n>N$:  :$ \left\vert \displaystyle \sum_{k \mathop = n}^m a_n e^{-\lambda_ks_0} e^{-\lambda_ks} \right\vert < \epsilon$ By Abel's Lemma: Formulation 2 we may write: {{begin-eqn}} {{eqn | l = \left\vert \sum_{k \mathop = n}^m a_k e^{-\lambda_ks_0} e^{-\lambda_ks} \right\vert       | r = \left\vert \sum_{k \mathop = n}^m \left(S\left(k,n\right) - S\left(k-1,n\right)\right) e^{-\lambda_ks} \right\vert }} {{eqn | r = \left\vert S\left(m,n\right)e^{-\lambda_ms} + \sum_{k \mathop = n}^{m-1} S\left(k,n\right) \left(e^{-\lambda_ks} - e^{-\lambda_{k+1}s}\right) \right\vert }} {{end-eqn}} Because $S\left(m,n\right)$ is the difference of partial sums of a convergent, and thus cauchy, sequence, its modulus, $\left\vert S\left(m,n\right) \right\vert$, is bounded, say by $Q$. Thus we have: {{begin-eqn}} {{eqn | l = \left\vert S\left(m,n\right)^{-\lambda_ms} + \sum_{k \mathop = n}^{m-1} S\left(k,n\right) \left(e^{-\lambda_ks} - e^{-\lambda_{k+1}s}\right) \right\vert       | o = \leq       | r = \left\vert S\left(m,n\right) e^{-\lambda_ms} \right\vert + \sum_{k \mathop = n}^{m-1} \left\vert S\left(k,n\right) \left(e^{-\lambda_ks} - e^{-\lambda_{k+1}s}\right) \right\vert }} {{eqn | o = \leq       | r = Qe^{-\lambda_m\sigma}+ Q \sum_{k \mathop = n}^{m-1} \left\vert e^{-\lambda_ks} - e^{-\lambda_{k+1}s} \right\vert }} {{end-eqn}} We see that: {{begin-eqn}} {{eqn | l = \left\vert e^{-\lambda_ks} - e^{-\lambda_{k+1}s} \right\vert       | r = \left\vert \int_{\lambda_k}^{\lambda_{k+1} } -se^{-xs} \rd x \right\vert }} {{eqn | o = \leq       | r = \int_{\lambda_k}^{\lambda_{k+1} } \left\vert-se^{-xs}\right\vert \rd x       | c = Modulus of Complex Integral }} {{eqn | r = \int_{\lambda_k}^{\lambda_{k+1} } \left\vert s \right\vert e^{-x\sigma} \rd x       | c =  }} {{eqn | r = \left\vert s \right\vert \int_{\lambda_k}^{\lambda_{k+1} } e^{-x\sigma} \rd x       | c =  }} {{eqn | r = \frac {\left\vert s \right\vert} {\sigma} \left( e^{-\lambda_k\sigma} - e^{-\lambda_{k+1}\sigma} \right)       | c =  }} {{end-eqn}} Thus we have: {{begin-eqn}} {{eqn | l = Qe^{-\lambda_m\sigma} + Q \sum_{k \mathop = n}^{m-1} \left\vert e^{-\lambda_ks} - e^{-\lambda_{k+1}s} \right\vert       | o = \leq        | r = Qe^{-\lambda_m\sigma} + Q \sum_{k \mathop = n}^{m-1} \frac {\left\vert s \right\vert} {\sigma} \left( e^{-\lambda_k\sigma} - e^{-\lambda_{k+1}\sigma} \right) }} {{eqn | r = Q e^{-\lambda_m\sigma} + Q \frac {\left\vert s \right\vert} {\sigma} \left( e^{-\lambda_n\sigma} - e^{-\lambda_{m}\sigma} \right)       | c = Telescoping Series }} {{end-eqn}} Because $\lambda_n$ tends to infinity, both summands tend to 0 as n goes to $\infty$ if $\sigma>0$. Thus we can pick $N$ large enough such that both terms are less than $\dfrac {\epsilon} {2}$ for $n,m>N$, giving us the desired result {{qed}} Category:General Dirichlet Series jze59yk1w21lck7aodxk0ikfizdrl14