Proving the Boundedness of the Sequence ( frac{2n-1}{3n^2} )

In the field of mathematics, proving the boundedness of a sequence is a crucial concept. The sequence in question is ( a_n frac{2n-1}{3n^2} ). This article aims to provide a detailed explanation of how to prove that this sequence is bounded and to calculate its limit as ( n ) approaches infinity.

Proof of Boundedness

To prove that the sequence ( a_n frac{2n-1}{3n^2} ) is bounded, we need to show that there exist real numbers ( M ) and ( m ) such that for all ( n geq 1 ), ( m leq a_n leq M ).

Upper Bound

We start by showing that ( a_n leq frac{1}{3} ) for all ( n geq 1 ).

( a_n frac{2n-1}{3n^2} ). ( a_n frac{2n-1}{3n^2} frac{6n-3}{9n^2} frac{6n^2 - 7n 3 - 7n 3}{9n^2} frac{6n^2 - 7n 3}{9n^2} - frac{7n - 3}{9n^2} ). Since ( frac{7n - 3}{9n^2} leq frac{1}{3} ) for all ( n geq 1 ) (as ( frac{7n - 3}{9n^2} ) approaches 0 as ( n ) increases), we have ( a_n leq frac{1}{3} ).

Thus, the upper bound ( M frac{1}{3} ).

Lower Bound

To find the lower bound, we need to ensure that ( a_n geq 0 ) for all ( n geq 1 ).

( a_n frac{2n-1}{3n^2} geq 0 ) if ( 2n - 1 geq 0 ), which is true for all ( n geq 1 ). For ( n geq 1 ), ( a_n ) is always positive and decreases as ( n ) increases.

Thus, the lower bound ( m 0 ).

Limit as ( n ) Approaches Infinity

To determine the limit of the sequence ( a_n ) as ( n ) approaches infinity, we follow these steps:

( a_n frac{2n-1}{3n^2} ). ( lim_{n to infty} a_n lim_{n to infty} frac{2n-1}{3n^2} lim_{n to infty} frac{2n/n - 1/n}{3n^2/n^2} lim_{n to infty} frac{2 - frac{1}{n}}{3} frac{2}{3} ).

Therefore, as ( n ) approaches infinity, ( a_n ) approaches (frac{2}{3}).

The Sequence ( a_n ) in Detail

The terms of the sequence ( a_n ) for ( n 1, 2, 3, ldots ) are given by:

For ( n 1 ): ( a_1 frac{2(1)-1}{3(1)^2} frac{1}{3} ). For ( n 2 ): ( a_2 frac{2(2)-1}{3(2)^2} frac{3}{12} frac{1}{4} ). For ( n 3 ): ( a_3 frac{2(3)-1}{3(3)^2} frac{5}{27} ). For ( n 4 ): ( a_4 frac{2(4)-1}{3(4)^2} frac{7}{48} ). For ( n 5 ): ( a_5 frac{2(5)-1}{3(5)^2} frac{9}{75} frac{3}{25} ).

Notice that the sequence ( a_n ) is bounded below by ( frac{1}{5} ) (since ( a_5 frac{3}{25} ) and ( frac{3}{25} geq frac{1}{5} )) and is strictly increasing to (frac{2}{3}).

Conclusion

In conclusion, the sequence ( a_n frac{2n-1}{3n^2} ) is bounded and its limit as ( n ) approaches infinity is (frac{2}{3}). The sequence is bounded below by ( frac{1}{5} ) and above by ( frac{1}{3} ), and it is strictly increasing throughout.