MATH 107 PRACTICE MIDTERM 1
Please work out each of the given
problems. Credit will be based on
the steps that you show towards the final answer.
Show your work.
PROBLEM 1 Please answer the following true or false. If false, explain why or provide a counter example. If true explain why.
PROBLEM 2 (25 Points)
Determine the Taylor series representation centered at x = 2 of
PROBLEM 3 (25 points)
Determine the radius and interval of convergence of
To find the radius of convergence, we use the ratio text
|x - 4| < 1/2
2x - 8 = 1 or 2x - 8 = -1
x = 9/2 or x = 7/2
Now test the endpoints. Plugging in x = 9/2 gives
which converges by the alternating series test.
Plugging in x = 7/2 gives
which diverges by either the integral test or the limit comparison test (comparing with the divergent harmonic series S1/n.
We conclude that the interval of convergence is
For a Normal
Distribution (Bell curve) the integral
P(0 < z < x)
This quantity is
largest when x is1.
PROBLEM 5 Determine if the following series converge or diverge. Explain which test(s) you are using and show all your work.
Hence by the limit test, the series diverges
We use the ratio test
Hence by the ratio test, the series converges
By the geometric series test, with r = -1/3, the series converges absolutely.