136.275, Assignment No. 2

October 17, 2003

The assignment is due Friday, October 24 , 2003 in class. Late assignments receive a mark zero.

1.a) Show that if the series and converges, then

must also converge. [3]

b) If converges, must and converge? Justify your statement. [3]

c) Give an example where and both diverge, but converges. [2]

2. Let with p a real number.

a) For p< -1 use the Integral Test to determine if the series converges .

( Show that you can use the Integral test for those p.) [5]

b) For p -1 use the limit comparison test to determine the convergence of . [3]

3. Determine if the following series converge or not. State which test you are using:

a) , [4]

b) , [3]

c) [4]

4. Find the radius of convergence and the interval of convergence for the series

. [6]

5. a) Find the Taylor series expansion of f(x)= sinx about x₀= and find the radius

of convergence of the series. [4]

b) Show that f(x) = sinx equals to theTaylor series from a) for all x in |R, by using the

Remainder Theorem. [4]

c) Can we expand lnx about x₀=0? Can lnx be represented by a power series about

x₀=2 on an interval with radius greater than 2? Explain. [3]

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