Show that the sequence { b n } : b n = 5 n 2 + 2 n − 1 3 n 2 − 14 converges

Question

Show that the sequence       {          b      n        }    :      b    n    =                    5                  n          2                +        2        n        −        1                    3                  n          2                −        14             converges

massetereqe · Accepted Answer

Just use brute force:      b          n      +      1        −      b    n    =            5              n        2            +      12      n      +      6              3              n        2            +      6      n      −      11        −            5              n        2            +      2      n      −      1              3              n        2            −      14        =  −            6              n        2            +      140      n      +      95              (      3              n        2            −      14      )      (      3              n        2            +      6      n      −      11      )      So it easy to see that       b          n      +      1        −      b    n    &amp;lt;  0 for   n  ≥  3Also       b    n    &amp;gt;  0 for the same   n  ≥  3, so       b    n   is bounded as well, thus Weierstraß theorem applies.

Show that the sequence { b n </msub> } </mrow> : b n

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