When the Ratio Test fails, i.e. when \lim_{n o\infty}\frac{u_n}{u_{n+1}}=1 , we can use the Raabe's Test. The test states that the associated series converges/diverges according to \lim_{n o\infty}n(\frac{u_n}{u_{n+1}}-1) is greater/less than 1. Which of the following series fails the Ratio Test but passes the Raabe's Test for convergence/divergence?

Question

When the Ratio Test fails, i.e. when   \lim_{n	o\infty}\frac{u_n}{u_{n+1}}=1 , we can use the Raabe's Test. The test states that the associated series converges/diverges according to   \lim_{n	o\infty}n(\frac{u_n}{u_{n+1}}-1)  is greater/less than 1. Which of the following series fails the Ratio Test but passes the Raabe's Test for convergence/divergence?

Accepted Answer

\sum_{n=1}^{\infty}\frac{1}{n^2(n+1)^2}

Accepted Answer

\sum_{n=1}^{\infty}\frac{1}{n^2}

Accepted Answer

\sum_{n=1}^{\infty}\frac{1}{n(n+1)}

Answer

\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}

Answer

\sum_{n=1}^{\infty}\frac{1}{n}

Answer

\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}\sqrt{n+1}}

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When the Ratio Test fails, i.e. when \lim_{n\to\infty}\frac{u_n}{u_{n+1}}=1 , ...

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