Ma1502Te2Solns-HSpring2010

# Ma1502Te2Solns-HSpring2010 - MATH1502 Calculus II TEST 2 H...

This preview shows pages 1–4. Sign up to view the full content.

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

View Full Document

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: MATH1502 - Calculus II TEST 2 - H Group - March 4 - Spring 2010 Question Score Maximum 1 7 2 7 3 18 4 9 5 10 6 7 Total 58 Name _____________________________ Group (e.g. G1 or J1) _____________________ Student Number ____________________ Teaching Assistant ____________________ Answer all questions. There are 58 marks on the paper. 100% = 50 marks No &cheatsheets¡or calculators are allowed. Question 1 Let b be a number such that < b < 1 : Find the sum of the series 1 X k =1 1 ( k & b ) ( k & b + 1) : (6 marks) Deduce the value of 1 X k =1 1 & k & 1 3 ¡& k + 2 3 ¡ : (1 mark) Solution We use partial fractions: 1 ( k & b ) ( k & b + 1) = 1 k & b & 1 k & b + 1 : 1 The n th partial sum is s n = n X k =1 1 ( k & b ) ( k & b + 1) = n X k =1 & 1 k & b & 1 k & b + 1 ¡ = & 1 1 & b & 1 2 & b ¡ + & 1 2 & b & 1 3 & b ¡ + & 1 3 & b & 1 4 & b ¡ + ::: + & 1 n & 1 & b & 1 n & b ¡ + & 1 n & b & 1 n & b + 1 ¡ = 1 1 & b & 1 n & b + 1 : So lim n !1 s n = lim n !1 ¢ 1 1 & b & 1 n & b + 1 £ = 1 1 & b : Thus 1 X k =1 1 ( k & b ) ( k & b + 1) = 1 1 & b : (6 marks) In particular, choosing b = 1 3 ; 1 X k =1 1 ¤ k & 1 3 ¥¤ k + 2 3 ¥ = 1 1 & 1 3 = 3 2 : (1 mark) 2 Question 2 Test the following series for convergence or divergence: 1 X k =2 ( k 2 + 1) & k 1 = 3 + ( & 1) k ¡ k ( k + 1) ( k + 3) : (7 marks) Solution We see that for very large k , the series terms behave roughly like k 2 k 1 = 3 k ( k ) ( k ) = k & 2 = 3 : Since P k & 2 3 diverges ( p & series with p = 2 3 ) , this suggests we use the limit comparison test with a k = ( k 2 + 1) & k 1 = 3 + ( & 1) k ¡ k ( k + 1) ( k...
View Full Document

{[ snackBarMessage ]}

### Page1 / 10

Ma1502Te2Solns-HSpring2010 - MATH1502 Calculus II TEST 2 H...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online