Example of a function for which $\int_1^\infty f(x)\;dx$ diverges and $\sum_{1}^\infty f(n)$ converges

Question

I recently came across this one:

A function which is continuous and positive for $x \geq 1$ and such that $\int_1^\infty f(x)\;dx$ diverges and $\sum_{1}^\infty f(n)$ converges

The author gives this answer:

I understand what the function $g$ is .

My questions are:

• Is there anything special about the function $g$ here?
• Why define $f$ by adding $g$ to $\frac{1}{x^2}$ ?
• Why $f$ is continuous?
• Is there any other simple functions $g$ instead of the given one to make $f$ easier?

For my second question, I think we know $\sum \frac{1}{n^2} < \infty $ and $g(n)=0$ for integer points. So we define like this. Is this correct?

For my third question, I think $g$ is continuous and $\frac{1}{x^2}$ is continuous, so their sum is continuous. Is there any reason apart from this?

Any help must be appreciated and thanks in advance!

Answer 1

Let consider for example

$$g(x)=\sin^2(2\pi x) \implies g(n)=\sin^2 (2\pi n)=0\quad \forall n$$

The $1/x^2$ term has been added in order to have $f(n)>0$.

The continuity was considered in order to exclude simple example for $f$ as for example $f(x)=1$ $\forall x\not \in \mathbb{Z}$ and $f(x)=1/x^2$ otherwise.

Answer 2

As little constraints are imposed on $f$, you can very well choose a smooth function which is zero at integer values and positive elsewhere, such as $\sin^2(\pi x)$.

If the function must be strictly positive, add a term with sufficient decrease speed to guarantee convergence of the series, say $e^{-x}$ (the book chose $x^{-2}$).