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Q.
The sum of infinity of the series $\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+\frac{9}{400}+\ldots \ldots \infty$ is
Sequences and Series
Solution:
$ S _1=3,5,7,9, \ldots \ldots $
$T _{ m }=2 m +1$
$S _2=4,36,144,400 \ldots \ldots$
$T _{ p }=( p \times( p +1))^2 $
$T _{ n }=\frac{2 n +1+ n ^2- n ^2}{ n ^2( n +1)^2}=\frac{( n +1)^2- n ^2}{( n +1)^2 n ^2}=\frac{1}{ n ^2}-\frac{1}{( n +1)^2}$
$T _1=\frac{1}{1^2}-\frac{1}{2^2} $
$T _2=\frac{1}{2^2}-\frac{1}{3^2}$
$\Lambda = T _1+ T _2 \ldots \ldots \infty=1 .$