Question

Let $\sum_{n=1}^\infty \frac{1}{n^2}$ and $\sum_{n=1}^\infty \frac{1}{n^{\frac{1}{2}}}$ be series, then}

• A. both are convergent
• B. both are divergent
• C. $\sum_{n=1}^\infty \frac{1}{n^2}$ is convergent and $\sum_{n=1}^\infty \frac{1}{n^{\frac{1}{2}}}$ is divergent
• D. $\sum_{n=1}^\infty \frac{1}{n^2}$ is divergent and $\sum_{n=1}^\infty \frac{1}{n^{\frac{1}{2}}}$ is convergent

Hint:

Remember the threshold value for the p-series test:
- Any power \(\mathbf{p}\) strictly greater than 1 converges.
- Even a value like \(p = 1.0001\) converges.
- Any power \(\mathbf{p}\) less than or equal to 1 (including the harmonic series where \(p=1\)) diverges.

Answer 1

The Correct Option is C