Question11. 1) 10] Consider the infinite series: $$\sum_{n=1}^{\infty} a_{n}$$
With the partial sums, $S_{n}$. a) If $\lim _{n \rightarrow \infty} a_{n}=\sqrt{\pi}$, what can we say about the series convergence or divergence? b) If $\lim _{n \rightarrow \infty} S_{n}=\sqrt{\pi}$, what can we say about the series convergence or divergence?

Studdy Solution
Since $\lim_{n \rightarrow \infty} S_{n} = \sqrt{\pi}$, the series $\sum_{n=1}^{\infty} a_{n}$ converges to $\sqrt{\pi}$.
a) The series $\sum_{n=1}^{\infty} a_{n}$ diverges because $\lim_{n \rightarrow \infty} a_{n} \neq 0$.
b) The series $\sum_{n=1}^{\infty} a_{n}$ converges to $\sqrt{\pi}$ because $\lim_{n \rightarrow \infty} S_{n} = \sqrt{\pi}$.