I have the following problem:
Let $a_n$, $n\in\mathbb{N}$ be sequence and let $b_n=a_n+a_{n+1}+a_{n+2}$
Prove that if $\sum_{n=1}^{\infty}a_n$ converges then $\sum_{n=1}^{\infty}b_n$ converges
My attempt:
Let $\sum_{n=1}^{\infty}a_n=a$
$\sum_{n=1}^{\infty}b_n=\sum_{n=1}^{\infty}a_n+a_{n+1}+a_{n+2}$=
$\sum_{n=1}^{\infty}a_n+\sum_{n=1}^{\infty}a_{n+1}+\sum_{n=1}^{\infty}a_{n+2}$=
$\sum_{n=1}^{\infty}a_n$+$\sum_{n=1}^{\infty}a_n-a_1$+$\sum_{n=1}^{\infty}a_n-a_1-a_2$=
$\sum_{n=1}^{\infty}3a_n-2a_1-a_2$
From the linearity of series we know that $\sum_{n=1}^{\infty}3a_n=3a$
And the series convergence isn't affected by a change in finite number of elements of the sum
So $\sum_{n=1}^{\infty}b_n$ converges
Is this any good? Is it sufficient?