im trying to prove to following:
suppose $a_n , b_n $ sequences such that $a_n , b_n>0, a_n ,b_n\rightarrow0$ than if $\sum^{\infty}_{n=1} a_n$ converges and $\sum^{\infty}_{n=1} b_n$ diverges and $c_n = a_n + b_n$ then$\sum^{\infty}_{n=1} c_n$ diverges.
i proved by contradiction and assumed that it converges. then defined $d_n = c_n - a_n$ from my assumption i can say that $\sum^{\infty}_{n=1} d_n $ converges because its a sum of 2 converging series. then $$\sum^{\infty}_{n=1} d_n =\sum^{\infty}_{n=1} c_n -a_n \overset{\ast}{=} \sum^{\infty}_{n=1} a_n + b_n -a_n = \sum^{\infty}_{n=1} b_n$$ in contradicion to the fact that $\sum^{\infty}_{n=1} b_n$ diverges.
but im not really sure about the $\ast$ step. thank you!