# 14-15 - SOLUTIONS FOR HOMEWORK 7 14.1. (a) an = n4 /2n . We...

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

SOLUTIONS FOR HOMEWORK 7 14.1. (a) a n = n 4 / 2 n . We know that lim n 1 /n = 1 (Example 9.7). There- fore, lim | a n | 1 /n = 1 / 2 < 1. By Root Test, the series converges. (b) a n = 2 n /n !. lim( n !) 1 /n = + (proved in class), hence lim | a n | 1 /n = 0. By Root Test, the series converges. Alternatively , note that | a n +1 | / | a n | = 2 / ( n + 1), hence lim | a n +1 | / | a n | = 0. By Ratio Test, our series converges. (e) 0 a n = cos 2 n/n 2 b n = 1 /n 2 . b n converges (Integral Test), hence, by Comparison Test, a n converges. 14.4. (a) a n = 1 / ( n + ( 1) n ) 2 . The series n a n converges. For n 2, 0 a n b n = 1 / ( n 1) 2 , hence | a n | ≤ b n . However, n b n converges (Example 2 from p. 69), hence, by Comparison Test, n a n also converges. (b) a n = n + 1 n . The series n a n diverges to + . To see this, consider the sequence of partial sums: s n = n k =1 a k = n k =1 ( k + 1 k ) = n + 1 1 (this is a “telescopic sum”). Thus, lim s n = + . Alternatively , one could observe that a n = ( n + 1 n ) · n + 1 + n n + 1 + n = 1 n + 1 + n > 1 2 n + 1 = b n . However, n b n diverges, by Integral Test. (c) [ This is a bonus problem – very little partial credit is given ] a n = n ! /n n . The series n a n converges. For instance, one can apply Root Test. Note ±rst (it was probably shown in the discussion) that, for any n N , (1) n ! ( n/ 2) n/ 2 1 · n n/ 2+1 = 2 n n 2 n . Indeed, consider ±rst the case of n even, that is, n = 2 m . Then n ! = ( 1 · 2 · . . . · m ) · ( ( m + 1) · . . . · n ) . Then 1

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.

## 14-15 - SOLUTIONS FOR HOMEWORK 7 14.1. (a) an = n4 /2n . We...

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

View Full Document
Ask a homework question - tutors are online