# convergence of sum of 2 series

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!

Why so complicated ? You define $$d_n = c_n - a_n$$. Are you not aware of $$d_n=b_n$$ ?

Suppose that $$\sum^{\infty}_{n=1} c_n$$ converges, then $$\sum^{\infty}_{n=1} (c_n-a_n)$$ converges. This gives the convergence of $$\sum^{\infty}_{n=1} b_n.$$ A contradiction !

• You assume that they know that if $\sum x_n$ converges and $\sum y_n$ converges then $\sum x_n+y_n$ converges. My taste is that these questions are equivalent so it's strange to assume one to prove the other – Yanko May 14 '20 at 10:35
• @Yanko i can assume that, i just wanted to know that it is correct to subtract $a_n$ from $c_n$ – Elad Elmakias May 14 '20 at 10:37
• @EladElmakias Alright then – Yanko May 14 '20 at 10:38

$$\sum_{n=1}^\infty c_n-a_n = \lim_{N\rightarrow\infty} \sum_{n=1}^N c_n - a_n = \lim_{N\rightarrow\infty} \sum_{n=1}^N a_n+b_n-a_n = \sum_{n=1}^\infty a_n+b_n-a_n$$

$$a_n,b_n >0$$;

$$b_n \lt a_n+b_n=:c_n$$;

$$\sum b_n$$ diverges implies $$\sum c_n$$ diverges (Comparison test).