{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw14solutions - Math 521 Advanced Calculus I Instructor J...

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

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: Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: March 1, 2010 Assignment 14 1. Show that (a) ∞ X n =0 1 ( n + 1)( n + 2) = 1, (b) ∞ X n =0 1 ( α + n )( α + n + 1) = 1 α > , if α > 0, (c) ∞ X n =1 1 n ( n + 1)( n + 2) = 1 4 . To each of these, we apply partial fractions. For (a), we have 1 ( n + 1)( n + 2) = 1 n + 1- 1 n + 2 . Thus, s k = ((1- (1 / 2)) + ((1 / 2)- (1 / 3)) + ((1 / 3)- (1 / 4)) + ... + ((1 /k )- (1 /k + 1)) + ((1 /k + 1)- (1 /k + 2)) = 1- 1 k + 2 . Since s k → 1, part (a) follows. Part (b) follows similarly by noting (via partial fractions) that 1 ( n + α )( α + n + 1) = 1 n + α- 1 n + α + 1 and as such s k = 1 α- 1 k + α + 1 → 1 α . For part (c), we have 1 n ( n + 1)( n + 2) = 1 / 2 n- 1 n + 1 + 1 / 2 n + 2 and s k = (1 / 4)- 1 / 2 k + 1 + 1 / 2 k + 2 → 1 / 4 . 1 2. Let ∑ ∞ n =1 a ( n ) be such that { a ( n ) } is a decreasing sequence of strictly positive numbers. If s ( n ) denotes the n th partial sum, show (by grouping the terms in...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

hw14solutions - Math 521 Advanced Calculus I Instructor J...

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

View Full Document
Ask a homework question - tutors are online