Practice Third Midterm Examination, Turn in for April 28 HW. Math 172 Spring 2010, Wilkerson Section.
I. For each of the following infinite series, prove that it converges or that it diverges. Your explanation should include the name of the test used to determine convergence or divergence.
P
a) ∞
n=0 (5n + 1)/(2n + 3).
P∞
b) n=1 n(1/3)n .
P
c) ∞
n=2 1/(nln(n)) .
P∞
d) n=3 1/(n2 + 2n + 1).
II. For each of the following power series determine the radius of convergence, using the ratio
test.
a)
b)
P∞
n
n=0 5n(x/5) .
P∞
n=0
3n/2 (x − 1)3n .
III. For each of these functions, give the Taylor series about x = 0, and the radius of convergencefor each. You don’t have to develop the series from scratch.
a) 1/(1 − x3 ).
b) e2x .
c) sin(3x).
IV. Use the binomial theorem to give the expansion of (1 + x)1/2 about x = 0. Use the first two
non-zero terms to approximate (1/2)1/2 i, that is, evaluate T1 (−1/2), where T1 is the Taylor polynomial of degree 1. Now assume we have shown that for f (x) = (1 + x)1/2 , max[−.6,.6] |f ′′ (x)| ≤ M, for
some constant M. Use the Taylor remainder formula to estimate the error in your approximation
of (1/2)1/2 . It will depend on M.
V. Let f (x) = 1 + 2x + 3x2 + 4x3 + 5x4 + 6x5 . Compute the Taylor polynomial iT3 (x) of order 3 for f (x) about x = 1. Use it to approximate f (1.1).
VI. For each of these power series, do 1) differentiate and 2) integrate. If possible, tell what
1
functions these power series represent.
P
n+1
/(n + 1) .
a) ∞
n=0 x
P∞ n
b) n=0 x /n!.
2