Limit of a oscillating series and a denominator that goes to infinity

Question

So I have a formula of an average which is $\frac{\Sigma^n_{k =0}(-1)^k\times2}{n+1}$. What would be the limit as n $\rightarrow \infty$.

I know the series in the numerator is divergent and oscillating between 0 and 2. But, if the denominator goes to infinity, would the expression above still converge to zero?

Answer 1

Hint: can you produce upper and lower bounds for the numerator?

Answer 2

Let $$S_n=\frac{\sum_{k=0}^n(-1)^k2}{n+1}$$

then

$$S_{2n+1}=0\;\;\text{ and } \;S_{2n}=\frac{2}{2n+1}$$

$$\lim_{n\to\infty}S_{2n}=\lim_{n\to\infty}S_{2n+1}=0$$

$$\implies S_n\to 0$$