Some of the Sums

Let $S^{\infty}(k) = \displaystyle\sum_{n=1}^{\infty}(k)^{-n}$ . It is well know that our sum up to some term $n$ is evaluated by $S^{-n}(k) = \dfrac{a(1-k^{-n})}{1-k^{-1}}$ where $a$ is the first term of the series in or case $\dfrac{1}{k}$ . Now taking $\displaystyle\lim_{n\to \infty}(S^{-n}(k)) = \dfrac{\frac{1}{k}}{\frac{k - 1}{k}} = \dfrac{1}{k-1}$ .

Therefore our sum turns into $S = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{5} + \dfrac{1}{6} + \cdots$ or the Harmonic Series which $\boxed{\text{Diverges}}$ .