3. Direct & Limit Comparison Tests

Click for Solution

More notes on (Example 10 d):

(option 1) Apply ln to both sides,

ln ((ln n)^(ln n))= ln n (ln(ln n)) > 2 ln n = ln (n^2),

so, (ln n)^(ln n) > n^2

and 1/[ (ln n)^(ln n) ] < 1/(n^2) = b_n

(option 2) Observe that

(ln n )^(ln n) = n^(ln(ln n))

(this equality can be obtained by Applying ln to both sides, we have

ln ((ln n)^(ln n))= ln (n^(ln(ln n)))

and ln n (ln(ln n)) = ln(ln n) (ln n) )

Since n^(ln(ln n)) > n^2

we have (ln n )^(ln n) = n^(ln(ln n)) > n^2

hence 1/[ (ln n)^(ln n) ] < 1/(n^2) = b_n