TITLE: If series is divergent will a constant also keep it diverging?
QUESTION [5 upvotes]: If $\sum_{n=0}^{\infty} b_n$ diverges, and $c \in \mathbb{R}$ then does $\sum_{n=0}^{\infty} cb_n$ diverge?

My answer: NO.
Let $c=0$, then the sum is $\sum_{n=0}^{\infty} 0 = 0$.
True conclusion?

REPLY [2 votes]: Your answer is correct. However, if $c\neq 0$, then the sum does, indeed, diverge.
This is because if $\sum_{n=0}^\infty b_n$ converges, then the sum $\sum_{n=0}^\infty a\cdot b_n$ also converges for any constant $a\in\mathbb R$. So, if the right side would converge and $c\neq 0$, you can multiply the right sum by $\frac1c$ and get that the left side must converge. (therefore proving $\neg B\implies \neg A$ which also proves $A\implies B$).

REPLY [2 votes]: Your conclusion is true. Now what if $c\in\mathbb{R}\backslash\{0\}$? The answer will be YES! $0$ is an exception. In fact you'll even have equivalence when $c\neq 0$: $\sum_{n=0}^\infty b_n$ converges if and only if $\sum_{n=0}^\infty cb_n$ converges.