Math Book Cambodia: Find the limit of `S_n=2/(1.3)+2/(3.5)+2/(5.3)+.......+2/((2n+1)(2n+3))`

Monday, July 12, 2021

Solution

We can see that general term of this sequence is:

$2/((2k+1)(2k+3))=1/(2k+1)-1/(2k+3)$

We will replace that value of $k=0,1,2,....,n$

$k=1\leftrightarrow2/(1.3)=1-1/3$

$k=2\leftrightarrow2/(3.5)=1/3-1/5$

$k=3\leftrightarrow2/(5.7)=1/5-1/7$

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$k=n\leftrightarrow2/((2n+1)(2n+3))=1/(2n+1)-1/(2n+3)$

Sum of side to side:

Then, $S_n=1-1/(2n+3)$

When $n$ to infinity then $1/(2n+3)\rightarrow0$

Hence, Limit of $S_n$ is $1$ .

Solution by Thin Sokkean

Math Book Cambodia