{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# aC - (i ∞ ∑ k = 1 k 2 k(ii ∞ ∑ k = k k 1 3 k(iii...

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

University of Toronto, MAT 137Y, 2008–2009 Problem Set Supplement #3 Not to be handed in. This assignment will not count towards your course mark. The following exercises cover new material which you are responsible for the ﬁnal exam. 1. SHE Section 12.5: 4, 6, 12, 16, 28, 32, 46. 2. SHE Section 12.8: 6, 8, 10, 14, 16, 26, 46. 3. SHE Section 12.9: 2, 6, 14, 28, 42. 4. Find the radius of convergence for the following power series. (i) k = 1 k k k ! x k . (ii) k = 1 k k ( k ! ) 2 x k . (iii) k = 1 k k ( k ! ) 3 / 2 x k . (iv) k = 1 ( k ! ) 1 / k x k . 5. Starting with the formula 1 1 - x = 1 + x + x 2 + ··· , show that x d dx ± x d dx ± 1 1 - x ²² = x + 2 2 x 2 + 3 2 x 2 + ··· , and hence derive a formula for k = 1 k 2 x k . 6. Let { a n } be the Fibonacci sequence deﬁned by a 0 = a 1 = 1 and a n + 2 = a n + 1 + a n for n 0. (a) Prove that a n 2 n . (b) Let f ( x ) = n = 0 a n x n . Show that this power series has radius of convergence R 1 2 . (c) By computing f ( x ) - xf ( x ) - x 2 f ( x ) , show that f ( x ) = 1 / ( 1 - x - x 2 ) . (d) Deduce that a n = f ( n ) ( 0 ) n ! . 7. Evaluate the following series by converting each one to a power series.

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: (i) ∞ ∑ k = 1 k 2 k . (ii) ∞ ∑ k = k ( k + 1 ) 3 k . (iii) ∞ ∑ k = k 2 2 k . (iv) ∞ ∑ k = 1 2 k k ( k + 1 ) . 8. Find a power series expansion for f ( x ) = Z x e t 2 dt valid for all x . 9. Suppose a = 1, a 1 = 1, and a n + 2 = a n + 1 + 6 a n for n ≥ 0. (a) Find a 2 , a 3 , a 4 . (b) Prove that a n ≤ 6 n . (c) Let f ( x ) = ∞ ∑ n = a n x n . Show that this power series has radius of convergence R ≥ 1 6 . (d) Show that f ( x ) = 1 / ( 1-x-6 x 2 ) . (Hint: This is similar to a previous exercise.) (e) Use partial fractions to write f ( x ) = A 1-3 x + B 1 + 2 x for some constants A and B . (f) Use the power series expansions of 1 1-3 x and 1 1 + 2 x to show that a n = 3 5 · 3 n + 2 5 · (-2 ) n ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

aC - (i ∞ ∑ k = 1 k 2 k(ii ∞ ∑ k = k k 1 3 k(iii...

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

View Full Document
Ask a homework question - tutors are online