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

Question

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.

Answer 1

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 $k>0.$