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Calculus I
Sequence Theorems
Stolz Theorem
To find limits of indeterminate expressions x n y n \frac{{{x}_{{n}}}}{{{y}_{{n}}}} y n x n of type ∞ ∞ \frac{{\infty}}{{\infty}} ∞ ∞ often can be useful following theorem.
Stolz Theorem. Suppose that sequence y n → + ∞ {y}_{{n}}\to+\infty y n → + ∞ and starting from some number with increasing of n {n} n also increases y n {y}_{{n}} y n (in other words if m > n {m}>{n} m > n then y m > y n {y}_{{m}}>{y}_{{n}} y m > y n ). Then lim x n y n = lim x n − x n − 1 y n − y n − 1 \lim\frac{{{x}_{{n}}}}{{{y}_{{n}}}}=\lim\frac{{{x}_{{n}}-{x}_{{{n}-{1}}}}}{{{y}_{{n}}-{y}_{{{n}-{1}}}}} lim y n x n = lim y n − y n − 1 x n − x n − 1 if limit of the expression on the right side exists (finite or infinite).
Example 1 . Find lim a n n \lim\frac{{{{a}}^{{n}}}}{{n}} lim n a n where a > 1 {a}>{1} a > 1 .
Let x n = a n {x}_{{n}}={{a}}^{{n}} x n = a n and y n = n {y}_{{n}}={n} y n = n then y n {y}_{{n}} y n is increasing and y n → ∞ {y}_{{n}}\to\infty y n → ∞ therefore we can apply Stolz Theorem.
lim a n n = lim x n y n = lim x n − x n − 1 y n − y n − 1 = lim a n − a n − 1 n − ( n − 1 ) = lim ( a n − a n − 1 ) = \lim\frac{{{{a}}^{{n}}}}{{n}}=\lim\frac{{{x}_{{n}}}}{{{y}_{{n}}}}=\lim\frac{{{x}_{{n}}-{x}_{{{n}-{1}}}}}{{{y}_{{n}}-{y}_{{{n}-{1}}}}}=\lim\frac{{{{a}}^{{n}}-{{a}}^{{{n}-{1}}}}}{{{n}-{\left({n}-{1}\right)}}}=\lim{\left({{a}}^{{n}}-{{a}}^{{{n}-{1}}}\right)}= lim n a n = lim y n x n = lim y n − y n − 1 x n − x n − 1 = lim n − ( n − 1 ) a n − a n − 1 = lim ( a n − a n − 1 ) =
= lim a n ( 1 − 1 a ) = + ∞ =\lim{{a}}^{{n}}{\left({1}-\frac{{1}}{{a}}\right)}=+\infty = lim a n ( 1 − a 1 ) = + ∞ .
Example 2 . If lim a n = A \lim{a}_{{n}}={A} lim a n = A find lim a 1 + a 2 + . . + a n n \lim\frac{{{a}_{{1}}+{a}_{{2}}+..+{a}_{{n}}}}{{n}} lim n a 1 + a 2 + .. + a n (arithmetic mean of first n {n} n values of sequence a n {a}_{{n}} a n ).
Let x n = a 1 + a 2 + … + a n {x}_{{n}}={a}_{{1}}+{a}_{{2}}+\ldots+{a}_{{n}} x n = a 1 + a 2 + … + a n and y n = n {y}_{{n}}={n} y n = n then y n {y}_{{n}} y n is increasing and y n → ∞ {y}_{{n}}\to\infty y n → ∞ therefore we can apply Stolz Theorem.
lim a 1 + a 2 + … + a n n = lim x n y n = lim x n − x n − 1 y n − y n − 1 = \lim\frac{{{a}_{{1}}+{a}_{{2}}+\ldots+{a}_{{n}}}}{{n}}=\lim\frac{{{x}_{{n}}}}{{{y}_{{n}}}}=\lim\frac{{{x}_{{n}}-{x}_{{{n}-{1}}}}}{{{y}_{{n}}-{y}_{{{n}-{1}}}}}= lim n a 1 + a 2 + … + a n = lim y n x n = lim y n − y n − 1 x n − x n − 1 =
= lim ( a 1 + a 2 + … + a n ) − ( a 1 + a 2 + … . + a n − 1 ) n − ( n − 1 ) = lim a n 1 = lim a n = A =\lim\frac{{{\left({a}_{{1}}+{a}_{{2}}+\ldots+{a}_{{n}}\right)}-{\left({a}_{{1}}+{a}_{{2}}+\ldots.+{a}_{{{n}-{1}}}\right)}}}{{{n}-{\left({n}-{1}\right)}}}=\lim\frac{{{a}_{{n}}}}{{1}}=\lim{a}_{{n}}={A} = lim n − ( n − 1 ) ( a 1 + a 2 + … + a n ) − ( a 1 + a 2 + … . + a n − 1 ) = lim 1 a n = lim a n = A .
For example, since n n → 1 {\sqrt[{{n}}]{{{n}}}}\to{1} n n → 1 then 1 + 2 + 3 3 + … + n n n → 1 \frac{{{1}+\sqrt{{{2}}}+{\sqrt[{{3}}]{{{3}}}}+\ldots+{\sqrt[{{n}}]{{{n}}}}}}{{n}}\to{1} n 1 + 2 + 3 3 + … + n n → 1 .