Answers and comments for Math 242 Homework 11.


#1. Final answer is

x/(1*1!) + x^2/(2*2!) + x^3/(3*3!) + ... + x^n/(n*n!) + ...


#2. Final answer is

0.2 - (0.2)^6/6 + (0.2)^11/11 - (0.2)^16/16 + ...

or

1/5 - 1/(6*5^6) + 1/(11*5^11) - 1/(16*5^16) + ...

Sum of the first two terms is

0.2 - 0.000064/6  =  0.2 - 0.0000106666  =  0.1999893333


#3. First three terms of Taylor series simplify to

1 + x/3 - x^2/9

Estimate of 1.06^(1/3) is

1 + 0.06/3 - (0.06)^2/9

= 1 + 0.02 - 0.0036/9

= 1 + 0.02 - 0.0004

= 1.0196

By way of comparison, Maple gives 1.019612822.


#4. Graph should be the same as the graph of y = 1/x,

EXCEPT x can only be between 0 and 1
(and y can only be between 1 and infinity).

Direction on curve: down and to the right.


#5. First, using partial fractions, we find

3/((x-2)(x+1)) = 1/(x-2) - 1/(x+1).

To write as a power series, it may help to write it as

(-1/2)[ 1 / (1 - x/2) ] - [ 1 / (1 + x) ]

In the final answer, the coefficient of x^n should be

(-1/2)*(1/2)^n - (-1)^n.