11. which of the following series diverge? i. ∑_{k = 1}^{∞} (6k + 1)/(12k - 1) ii. ∑_{k = 0}^{∞} 20/10^k…

11. which of the following series diverge? i. ∑_{k = 1}^{∞} (6k + 1)/(12k - 1) ii. ∑_{k = 0}^{∞} 20/10^k iii. ∑_{k = 0}^{∞} (1/4)^k a. i only b. ii only c. iii only d. i and ii e. ii and iii

11. which of the following series diverge? i. ∑_{k = 1}^{∞} (6k + 1)/(12k - 1) ii. ∑_{k = 0}^{∞} 20/10^k iii. ∑_{k = 0}^{∞} (1/4)^k a. i only b. ii only c. iii only d. i and ii e. ii and iii

Answer

Explanation:

Step1: Check series I

Use the limit - comparison test. Compare $\sum_{k = 1}^{\infty}\frac{6k + 1}{12k-1}$ with the harmonic - like series $\sum_{k = 1}^{\infty}\frac{1}{1}$. Calculate $\lim_{k\rightarrow\infty}\frac{\frac{6k + 1}{12k-1}}{\frac{1}{1}}=\lim_{k\rightarrow\infty}\frac{6k + 1}{12k-1}=\lim_{k\rightarrow\infty}\frac{6+\frac{1}{k}}{12-\frac{1}{k}}=\frac{6}{12}=\frac{1}{2}>0$. Since $\sum_{k = 1}^{\infty}1$ diverges, by the limit - comparison test, $\sum_{k = 1}^{\infty}\frac{6k + 1}{12k-1}$ diverges.

Step2: Check series II

This is a geometric series of the form $\sum_{k = 0}^{\infty}ar^{k}$, where $a = 20$ and $r=\frac{1}{10}$. For a geometric series $\sum_{k = 0}^{\infty}ar^{k}$, if $|r|<1$, the series converges. Here, $|r|=\left|\frac{1}{10}\right|=\frac{1}{10}<1$, so $\sum_{k = 0}^{\infty}\frac{20}{10^{k}}$ converges.

Step3: Check series III

This is also a geometric series with $a = 1$ and $r=\frac{1}{4}$. Since $|r|=\left|\frac{1}{4}\right|=\frac{1}{4}<1$, the series $\sum_{k = 0}^{\infty}\left(\frac{1}{4}\right)^{k}$ converges.

Answer:

A. I only