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

Accepted Solution

Answer:TrueStep-by-step explanation:We have the serie:[tex]\frac{1}{25}+ \frac{1}{36} + \frac{1}{49}+...[/tex]To test whether the series converges or diverges first we must find the rule of the seriesNote that:[tex]5^2 = 25\\\\6^2 = 36\\\\7^2 = 49[/tex]Then we can write the series as:[tex]\frac{1}{5^2}+ \frac{1}{6^2} + \frac{1}{7^2}+...[/tex]Then:[tex]\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}[/tex]The series that have the form:[tex]\sum_{n=1}^{\infty}\frac{1}{n^p}[/tex]are known as "p-series". This type of series converges whenever [tex]p > 1[/tex].In this case, [tex]p = 2[/tex] and [tex]2 > 1[/tex]. Then the series converges