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Q.
If $S_{n}=\displaystyle\sum_{r=1}^{n} \frac{2 r+1}{r^{4}+2 r^{3}+r^{2}}$, then $S_{10}$ is equal to
Sequences and Series
Solution:
$S _{ n }=\displaystyle\sum_{ r =1}^{ n } \frac{2 r +1}{ r ^{2}( r +1)}=\displaystyle\sum_{ r =1}^{ n }\left[\frac{1}{ r ^{2}}-\frac{1}{( r +1)^{2}}\right]$
$\therefore S _{10}=\left(\frac{1}{1}-\frac{1}{2^{2}}\right)+\left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right)+\ldots .+\left(\frac{1}{10^{2}}-\frac{1}{11^{2}}\right)$
$S _{10}=1-\frac{1}{121}=\frac{120}{121}$