AP Calculus BC / Series / Question No. 6

6. Which of the following series converge?

I.\[ \sum_{n=1}^{\infty}\frac{n+1}{2n+1} \] II. \[ \sum_{n=1}^{\infty}\frac{1}{4n^2} \] III. \[ \sum_{n=1}^{\infty}\frac{2^n}{n3^n} \]

I only

II only

III only

II and III only

Answer is: option4

II and III only

Solution:

I. Diverges: Fails the nth term test: \[ \lim_{n\to\infty}\frac{n+1}{2n+2}=\frac{1}{2}\ne0 \] II. Converges: \(p\)-series with \(p=2>1\) III. Converges: By Ratio Test \[ \left|\frac{a_{n+1}}{a_n}\right| = \frac{n}{n+1}\cdot\frac{2}{3} \to \frac{2}{3}<1

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