Answer to Question #175908 in Real Analysis for Nikhil

Question

The series

1/3+ 1/7+ 1/11 + 1/15+.....

is a convergent series.

True or false with full explanation

Answer 1

The series

1/3+ 1/7+ 1/11 + 1/15+.....

is a convergent series.

True or false with full explanation

Solution:

$\frac13+\frac17+\frac{1}{11}+\frac{1}{15}+...=\displaystyle\sum_{n=1}^\infty\frac{1}{4n-1}=\\\frac14\displaystyle\sum_{n=1}^\infty\frac{1}{n-\frac14}=\frac14\displaystyle\sum_{n=1}^\infty a_n$

Let's apply direct comparison test:

$a_n=\displaystyle\frac{1}{n-\frac14}>\frac{1}{n}$

$\displaystyle\sum_{n=1}^\infty\frac{1}{n}$ is the harmonic series that is the divergent infinite series.

If $\displaystyle\sum_{n=1}^\infty\frac{1}{n}$ is a divergent series and $a_n>\frac{1}{n}$ then $\displaystyle\sum_{n=1}^\infty a_n$ is also a divergent series.

Since

$\displaystyle\sum_{n=1}^\infty a_n$ is a divergent series

than

$\frac14\displaystyle\sum_{n=1}^\infty a_n=\frac13+\frac17+\frac{1}{11}+\frac{1}{15}+...$ is also a divergent series.

Answer: false.