I'm asked if the sum of $\cos\left(\,\pi k/2\,\right)\left[\,k/\left(\,k + 1000\,\right)\,\right]/\sqrt{\,k\,}\,$ from $k = 1$ to $n$ for $n = 1,2,3,4\ldots$ is convergent or not.
I don't know how to begin. Am I asked to find for which n, this serie converges ?. Thank you for your help and any indication you could give me...
The sum $s_n$ is a finite sum of $n$ terms, so of course it is just some number for each $n$. You are being asked to find if $s_n$ converges.
That's exactly what it means to ask if the infinite series $$\sum_{k=1}^{\infty}\cos\left(\tfrac{\pi}{2}k\right)\tfrac{k}{k+1000}\tfrac{1}{\sqrt{k}}$$ converges. The definition of convergence of an infinite series is that the sequence of partial sums converges. So they are asking if $$\lim\limits_{N\to\infty}\sum_{k=1}^{N}\cos\left(\tfrac{\pi}{2}k\right)\tfrac{k}{k+1000}\tfrac{1}{\sqrt{k}}$$ exists.