Key to Practice Midterm I

Problem 1

1. False,  the derivative of a power series may no longer converge at the endpoints.

2. True, use the DCT.  a_n < a_n + b_n < c_n and b_n < a_n + b_n < c_n 

Problem 2

Proof

Problem 3

R = 4/3, Interval:  [-10/3, 2/3)

Problem 4

1. S (-1)ⁿx^6.5n+3

2. S (-1)ⁿx^6.5n+4/(6.5n + 4) + C

3. Set  1^6.5n+10.5/(6.5n + 10.5) < .005   
   gives  1/(6.5n + 10.5) < .005    or    6.5n + 10.5 > 200
   6.5n > 189.5    or    n > 29.15  n = 30
   The sum seq on the calculator produces  0.19.

Problem 5

1. Converges by the GST (r = 2/e < 1)

2. Since S1/n² converges, the LCT shows that our series converges.

3. Converges by the Ratio Test

4. Converges conditionally by a combination of the AST and the Integral Test