Test the convergence of the series 1/3.7+1/4.9+1/5.11+… | Investigate convergence of the series 1/3.7+1/4.9+1/5.11+… | infinite series problems with solution solved problems in calculus

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Test the convergence of the series 1/3.7+1/4.9+1/5.11+… OR  Investigate convergence of the series 1/3.7+1/4.9+1/5.11+…

 

Solution Here 

 

u_n=\frac{1}{(n+2)(2n+5)}

 
 
 

Let v_n=\frac{1}{n^2} now consider  \frac{u_n}{v_n}=\frac{1}{(n+2)(2n+5)}n^2={\frac{n^2}{2n^2+9n+10}}

 
 

\Rightarrow \lim_{n\rightarrow \infty}\frac{u_n}{v_n}=\lim_{n\rightarrow \infty}\frac{n^2}{2n^2+9n+10}

 

\lim_{n\rightarrow \infty}\frac{1}{2+\frac{9}{n}+\frac{10}{n^2}}=\frac{1}{2}

 

(which is a finite and non zero number)

 

Hence by Limit form test , \sum_{n=0}^{\infty}u_n  and \sum_{n=0}^{\infty}v_n behave similarly.

 
\sum_{n=0}^{\infty}v_n=\sum_{n=0}^{\infty}\frac{1}{n^2} converges (as p=2 > 1)
 
\therefore \sum_{n=0}^{\infty}u_n=\sum_{n=0}^{\infty}\frac{1}{(n+2)(2n+5)} also converges.

Test The Following Series

\frac{1}{3}+\frac{1.2}{3.5}+\frac{1.2.3}{3.5.7}+...\infty

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Related  If Σ an and Σ bn are series of positive terms and lim an/bn=0 from n to +∞ show by example that Σ an may converge and Σ bn not converge solved problems in calculus

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