Convergence of series $\sum{(a_n+k)}$ where k is a positive number and $a_n$ is a positive sequence

To prove that the series $$\sum{(a_n+k)}$$ converges or diverges according as the series $$\sum{a_n}$$ converges or diverges, where $$a_n$$ is sequence of positive terms and k is any positive real number.

According to me a series $$\sum{a_n}$$ of positive terms such that $$\\lim_{n\rightarrow \infty} a_n \neq 0$$ diverges. So, if $$\sum a_n$$ converges then $$\\lim_{n\rightarrow \infty} a_n = 0$$. If we consider $$\lim_{n\rightarrow \infty} (a_n +k) =\lim_{n\rightarrow\infty} a_n+\lim_{n\rightarrow\infty}k\\ =\lim_{n\rightarrow\infty} a_n+ k\\ = 0+k\neq 0$$ Thus, series $$\sum (a_n+k)$$ diverges. Which is opposite of what I am supposed to prove.

Please point out mistakes if any and provide the proof of question.

• Your argument is correct. The assertion you are supposed to prove is false. Think about $a_n=0$ for all $n$ and $k=1$. Are you sure you stated the assertion correctly? Should "sum" be "limit"? Jan 31 '21 at 15:20
• No it was sum in the given question. Thank you!! Feb 1 '21 at 4:20

You've got the right idea. The result is false: $$\sum a_n+k$$ diverges regardless of whether $$\sum a_n$$ converges or diverges, given that $$k>0$$ and each $$a_n>0.$$
This is because $$a_n+k>k>0$$ for all $$n,$$ so it is impossible for $$a_n\to 0$$ as $$n\to\infty.$$
Perhaps a more straightforward way to see the divergence is by direct comparison to the series $$\sum_{n=1}^\infty k,$$ which clearly diverges to $$+\infty$$ for any $$k>0.$$