11.1.25-eg

# 11.1.25-eg - Solution#1 Since lim n →∞ 12 n 2 2 n 5 n 5...

This preview shows pages 1–2. Sign up to view the full content.

Example Consider the sequence { a n } where the nth term is given by a n = 12 n 2 + 2 n 5 n 5 + 10 n + 1 Determine if the sequence converges or diverges. If the sequence converges, ﬁnd its limit. Note that the nth term a n is the same as the value of the function f ( x ) = 12 x 2 + 2 x 5 x 5 + 10 x + 1 evaluated at the point x = n . So the points ( n,a n ) = ( n,f ( n )) lie on the graph of the function y = f ( x ) in the x - y plane, and moreover, lim n →∞ a n = lim n →∞ f ( n ) = lim x →∞ f ( x ) . Thus, any techniques that can be used for a limit of the form lim x →∞ f ( x ) can be used for the limit lim n →∞ a n of the sequence { a n } . For the function f ( x ) given above, we can ﬁnd lim x →∞ f ( x ) using l’Hˆ opital’s rule, and also by dividing numerator and denominator of f ( x ) by the largest power of x appearing in both numerator and denominator. This leads to two diﬀerent approaches to the problem of deter- mining the limit lim n →∞ a n of the sequence { a n }

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: . Solution #1: Since lim n →∞ 12 n 2 + 2 n 5 n 5 + 10 n + 1 is of the form “ ∞ ∞ ”, we may use l’Hopital’s rule (repeatedly) to ﬁnd lim n →∞ 12 n 2 + 2 n 5 n 5 + 10 n + 1 = lim n →∞ 24 n + 10 n 4 5 n 4 + 10 = lim n →∞ 24 + 40 n 3 20 n 3 = lim n →∞ 120 n 2 60 n 2 = lim n →∞ 240 n 120 n = lim n →∞ 240 120 = 2 , so that lim n →∞ a n = 2. Solution #2: The largest power in the numerator and denominator of a n is n 5 , so if we divide numerator and denominator by n 5 we ﬁnd lim n →∞ 12 n 2 + 2 n 5 n 5 + 10 n + 1 = lim n →∞ (12 n 2 + 2 n 5 ) ÷ n 5 ( n 5 + 10 n + 1) ÷ n 5 = lim n →∞ 12 n 3 + 2 1 + 10 n 4 + 1 n 5 = 0 + 2 1 + 0 + 0 = 2 , leading once again to the conclusion that lim n →∞ a n = 2. 2...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

11.1.25-eg - Solution#1 Since lim n →∞ 12 n 2 2 n 5 n 5...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online