This document contains 24 review questions about key concepts in chapter 11 of a calculus textbook, including: 1) The definition of convergence for sequences and using it to prove limits. 2) The definitions of convergence for series, absolute convergence, and conditional convergence. 3) Tests for determining if series converge or diverge, including the nth term test. 4) Power series, including radius of convergence, intervals where series converge absolutely or conditionally. 5) Taylor series and polynomials, Taylor's formula and remainder theorem, and using series to approximate functions and evaluate limits.

1. CHAPTER 11 REVIEW QUESTIONS
(1) State the − N definition of convergence of a sequence.
1
(2) Prove that limn→∞ 2n+3 = 0 using the above definition of convergence.
(3) What does it mean if we say the series an converges?
√
n
(4) Find the limit of the sequence an = n3 .
√
n
(5) Does the series n3 converge or diverge? Why?
6
(6) Find the limit of the sequence an = n .
5
6
(7) Does the series converge or diverge? Why?
5n
(8) Is it ever possible to conclude a series converges using the nth −term test?
(9) Determine whether the following series converge or diverge. Justify your
answer. If you use a theorem or test, state the name.
2
a.
3n + 1
2
b.
1 + en
nn
c.
(2n )2
(−1)n
d.
n2/3
(10) State the definition of absolute convergence and conditional convergence.
(11) Is it possible for a series to converge absolutely, but not converge? If yes,
give an example.
(12) Is it possible for a series to converge, but not converge absolutely? If yes,
give an example.
(13) Do the following series converge absolutely, conditionally or diverge?
(−1)n+1
a. √ 1/3 √
n n
cos(nπ)
b. √
n2 n
1
c.
(n + 1)(n + 2)(n + 3)
1

2. 2 CHAPTER 11 REVIEW QUESTIONS
(14) State the definition of a power series.
(15) For the following state (i) The radius and interval of convergence, (ii) for
which values of x does the series converge absolutely, (iii) for which values
of x does the series converge conditionally, (iv) for which values of x does
the series diverge.
∞
xn
a.
n=1
n
b. (−2)n (n + 1)(x − 1)n
∞
(x + 4)n
c.
n=1
n3n
(16) If the Taylor series generated by f (x) converges, does it necessarly converge
to f (x)?
(17) (a.) Find the Taylor series generated by f (x) = cos x at a = π/3.
(b). Show that the Taylor series converges to cos(x).
(18) Find the Taylor series generated by f (x) = 1/x2 at a = 1.
(19) Find the Taylor polynomial of order 3:
1
(a.) generated by f (x) = x+2 at a = 0
(b.) generated by f (x) = sin x at a = π/6.
(20) State Taylor’s Formula. What is the significance of this formula?
(21) State Taylor’s Remainder Estimation Theorem. What is the consequence
of this theorem?
(22) If cos x is approximated by its Taylor polynomial of order 3 for x such that
|x| < 0.5, what is the error estimate?
(23) (a.) Express cos(x2 ) as a power series.
1
(b.) Estimate 0
cos(x2 ) with an error of less than 0.001.
sin x − x − x3 /6
(24) Use series to evaluate lim . (Do not use L’Hospital’s Rule.)
x→0 x5