# If $\sum_{n=1}^{\infty}a_n$ converges then $\sum_{n=1}^{\infty}a_n+a_{n+1}+a_{n+2}$ converges

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?

• Your proof is correct. Here's a more general proof: if $\sum a_n, \sum b_n$ converge, then $\sum a_n + b_n$ converges and equals $\sum a_n + \sum b_n$. Use this and the fact that $\sum\limits_{n=1}^\infty a_{n+k}$ converges for every k if $\sum a_n$ exists. You basically used these statements within your proof. – James Yang Jan 22 at 20:32
• Yes your proof is correct – Mike Jan 22 at 20:35
• Please put in parentheses. – zhw. Jan 22 at 21:04