Question

(9CQ) The series 1/25+1/36+1/49... is convergent... True or False

Answer 1

Answer:

True

Step-by-step explanation:

We have the serie:

$\frac{1}{25}+ \frac{1}{36} + \frac{1}{49}+...$

To test whether the series converges or diverges first we must find the rule of the series

Note that:

$5^2 = 25\\\\6^2 = 36\\\\7^2 = 49$

Then we can write the series as:

$\frac{1}{5^2}+ \frac{1}{6^2} + \frac{1}{7^2}+...$

Then:

$\frac{1}{5^2}+ \frac{1}{6^2} + \frac{1}{7^2}+... = \sum_{n=5}^{\infty}\frac{1}{n^2}\\\\\sum_{n=5}^{\infty}\frac{1}{n^2} = \sum_{n=1}^{\infty}\frac{1}{(n+4)^2}$

The series that have the form:

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

are known as "p-series". This type of series converges whenever $p > 1$.

In this case, $p = 2$ and $2 > 1$. Then the series converges