Question

Let $r$ be any real number and $(a_n) = \frac{1}{n^r}$ be sequence of real number, then}

• A. $(a_n)$ is always convergent for any $r \in R$
• B. $(a_n)$ is convergent only for $r = 1$
• C. $(a_n)$ is convergent for any $r \geq 0$
• D. $(a_n)$ is divergent for any $r \in R$

Hint:

Do not confuse the convergence of a sequence with the convergence of an infinite series:
- The sequence \(\left(\frac{1}{n^r}\right)\) converges for all \(\mathbf{r \ge 0}\).
- The series \(\sum \frac{1}{n^r}\) converges only for \(\mathbf{r > 1}\) (p-series test).

Answer 1

The Correct Option is C