Question

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

Answer 1

True

Explanation:

We have the serie:

`(1)/(25)+ (1)/(36) + (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:

`(1)/(5^2)+ (1)/(6^2) + (1)/(7^2)+...`

Then:

`(1)/(5^2)+ (1)/(6^2) + (1)/(7^2)+... = \sum_(n=5)^(\infty)(1)/(n^2)\\\\\sum_(n=5)^(\infty)(1)/(n^2) = \sum_(n=1)^(\infty)(1)/((n+4)^2)`

The series that have the form:

`\sum_(n=1)^(\infty)(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