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

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

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