Show the series `T_n=(4n)/(n+3)` is monotonic if n is natural?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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It has to be shown that the series defined by `T_n = (4*n)/(n+3)` is monotonic if n is a natural numbers.

`T_(n+1) - T_n `

= `(4*(n+1))/(n+1+3) -(4*n)/(n+3)`

= `(4n + 4)/(n+4) - (4*n)/(n+3)`

= `((4n + 4)(n+3) - (4*n)(n+4))/((n+3)(n+4))`

= `(4n^2 + 12n + 4n + 12 - 4n^2 - 16n)/((n+3)(n+4))`

= `(12)/((n+3)(n+4))`

For any natural number n,  `(12)/((n+3)(n+4))` is positive.

This shows that the series with `T_n = (4*n)/(n+3)` is monotonic.

 

 

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