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Prove that the series \(\displaystyle{\sum_{k=2}^{\infty}\dfrac{\sin\,(k\,x)}{\ln\,k}}\) and the

series \(\displaystyle{\sum_{k=2}^{\infty}\dfrac{\sin\,(k\,x)}{k\,\ln\,k}}\) converge for each

\(\displaystyle{x\in\left[0,2\,\pi\right]}\).

Examine if the functions

\(\displaystyle{x\mapsto \sum_{k=2}^{\infty}\dfrac{\sin\,(k\,x)}{\ln\,k}\,,x\in\left[0,2\,\pi\right]}\)

and

\(\displaystyle{x\mapsto \sum_{k=2}^{\infty}\dfrac{\sin\,(k\,x)}{k\,\ln\,k}\,,x\in\left[0,2\,\pi\right]}\)

are continuous.

Greetings to all,

I think the exercise is quite easy and we will be basing the solution on the following facts:

1. The uniform limit of continuous functions is continuous.
2. The series $\sum a_n \sin nx$ converges uniformly throughout $\mathbb{R}$ if-f $n a_n \rightarrow 0$.

Now,

both serieses converge uniformly throughout $\mathbb{R}$ since observation $(2)$ holds and because the functions are continuous its uniform limit is a continuous function. Both these facts are quite well known results.

That's all folks.

The first function is not continuous.(no Lebesgue integrable)

The condition 2 is not true.
The sequence must be decreasing

S.F.Papadopoulos wrote:The first function is not continuous.(no Lebesgue integrable)

The condition 2 is not true.
The sequence must be decreasing

Oh sorry , my bad!!