If $\sum_{n=1}^\infty \frac{1}{f_n}$ diverges, does $\sum_{n=1}^\infty \frac{1}{1+f_n}$ also diverge?

Question

I have been trying to solve the following problem: Given $f(n): \mathbb{N} \rightarrow (0,\infty)$. If $\sum_{n=1}^\infty \frac{1}{f_n}$ diverges, does $\sum_{n=1}^\infty \frac{1}{1+f_n}$ also diverge?

I suspect that this is true, but I can't find a way of proofing this formally. The problem is that we don't know any other characteristics about $f_n$ other than that it is a positive function.

It might be the case that this is not true, but it would be nice in that case to see an example.

Thanks for your time

Answer 1

Suppose $\sum \frac{1}{1+f_{n}} < \infty$

Then $\lim_{n \to \infty} \frac{1}{1+f_{n}} = 0$

So $\lim_{n \to \infty} 1 + f_{n} = \infty$, thus $\lim_{n \to \infty} f_{n} = \infty$

Then there's exist $N \in \mathbb{N}$ such that $f_{n} > 1$ for all $n>N$

Then $\infty > \sum_{n = N+1}^{\infty} \frac{1}{1 + f_{n}} \geq \sum_{n = N+1}^{\infty} \frac{1}{f_{n} + f_{n}} = \frac{1}{2} \sum_{n = N+1}^{\infty}\frac{1}{f_{n}} = \infty$

That is a contradiction.

Then diverges too.

Answer 2

If $\lim_{n \to \infty} f_{n}$ DNE, then so also $\lim_{n \to \infty} \frac{1}{1 + f_{n}}$, and so the series diverges.

If $\lim_{n \to \infty} f_{n} = 0$, then $\lim_{n \to \infty} \frac{1}{1 + f_{n}} = 1$, and so the series diverges.

Otherwise, use the Limit Comparison Test with both of the series you have indicated, and you should be able to conclude divergence for these cases as the limit of the resulting quotient will be $1$.

Answer 3

Note that for $t \ge 1$ we have ${1 \over t} \le 2 { 1 \over t+t} \le {2 \over 1+t}$. In particular, if $f_n \ge 1$ we have ${1 \over f_n} \le 2 {1 \over 1+f_n}$.