※ 引述《CFE220 (五行之友)》之銘言：
: Find the limit
: x{[(x^2+1)^1/2]-[(x^3+1)^1/3]} when x → +∞
Let x^2+1=A, x^3+1=B.
Then the original statement = x(A^1/2-B^1/3)
= x(A^3-B^2)/{[A^1/2+B^1/3]*[A^2+A*B^2/3+B^4/3]}.
Replace A and B by x^2+1 and x^3+1 and divide the
numerator and the dominator by x^4,
then we get that the limit is 1/2.
: Determine the series
: ∞
: Σ {[(n^2+1)^1/2]-[(n^3+1)^1/3]}
: n=1
By the last problem we know that the summand is a big O of 1/n as n→∞,
so the series grows as fast as the harmonic series, which diverges.
Thus this series diverges.
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※ 編輯: bineapple 來自: 123.192.216.62 (07/08 00:36)