136.271

Assignment 3: Sections 9.5 (the part on differentiation and integration of series), 9.6 and 9.7 and 9.8

(Due March 21 in class)

Note: show your work; a naked final answer is not worth anything.

1. [6 marks] Find the sums of the following series.

(a) , .

(b) .

2. [5 marks] Find the Maclaurin series representation of the following functions.

(a)

(b)

3. [6 marks] Find the sum of the series.

(a)

(b)

(c)

4. [4 marks] Evaluate the following integrals as power series.

(a)

(b)

5. [4 marks] Show that the Lagrange remainder in the Taylor’s formula for the following functions tends to 0 as n tends to infinity, thus establishing that the functions are equal to their power series representations.

(a)

(b)