Homework 5: Due Nov 29th (Thursday)
(1) Evaluate the limits (note i, j, and k are the three basis vectors in R3 ):
(a) lim (t2 , 4t, 1/t)
t3
(b) lim (e2t i + ln(t + 1)j + 4k).
t0
r(t)
where r(t) = (sin(t), 1 cos(t), 2t).
t
Compute the following:
(a) r (t)
(1) (a) Let
(b) Let
(c) Let
(2) Suppose
Sample midterm 1
an = n n. What is a5 ?
br = 2ar . What is b2 ?
ck = ak ak+1 . Find a formula for ck .
that lim an = 4 and lim bn = 2. Determine
2
n
n
(a) lim an /bn
n
(b) lim sin(b2 )
n
n
(c) lim an2
n
(3) Let a0 =
Sample nal
(1) True or False
(a) There are convergent sequences that are not monotonic.
(b) If lim an = 10 then there is an integer N such that 9.9 < aN < 10.1.
n
(c) There is a sequence an so that a1 = a2 = a3 = = a10 = 1, but
lim an = 0.
n
(d) If 0 < li
Sample midterm 2
(1) The following sequences are convergent. Find their limits.
(a) an = nn+1 .
2
(b) an = n1/n .
(c) a1 = 1, and an+1 = 2 + an for n > 1.
(2) Which of these series converge and why.
n
(a)
log
n+1
n=1
(b)
n
3n
n=1
(c)
1
n(log n + 1)
n=1
1
Homework 5: Due Nov 20th (Tuesday)
x2n
Show that F (x) has innite radius of convergence.
2n n!
n=0
(2) Find the power series for f (x) = x log(1 + x ).
2
1
(3) Find the rst three terms of the Taylor series of f (x) = 1+tan x around 0.
(4) Evaluate
2n
(1)
Homework 5: Due Nov 1st (Thursday)
(1) Find the radius of convergence for
xn
.
2n
n=0
(2) Find the radius of convergence for
x2n
.
n!
n=0
nxn1 .
(3) Find the closed form for the function
n=0
(4) Find the power series expansion for the following functions:
Homework 4: Due Oct 18th (Thursday)
(1) Find the sum of the series i=1 i1 correct to 3 decimal places.
5
1
(2) How many terms of the series n=1 n log(n)2 would you need to add to fnd
its sum to within 0.01?
(3) Find the value of p for which the series n=1
Homework 2: Due Sept 25th
n
(1) Let b1 = 1 and bn+1 = b2 + b1 for all n > 1. Assuming that (bn ) is a
n
convergent sequence, nd its limit.
(2) Let an = (2n + 3n )1/n . Prove that
3 an 21/n 3
for all positive integers n. Calculate lim an .
n
(3) Find a for
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Homework 1: Due Sept 18th
What is a5 for the sequence an = n2 n?
Let a1 = 1 and an = 1 + a1 + a2 + + an1 for n > 1. Calculate the rst
four terms of the sequence (an ).
Let b1 = 2, b2 = 3 and bn = 2bn1 + bn2 for n > 2. Calcu
Homework 4: Due Oct 9th
(1) Which of the following series converge and why?
n1
(a)
n=1 n n
(b)
n=1
n=1
n=1
n=1
n=1
5n
6n +n
(sin n)2
n4 +n
n
en
1
(n!)2
1
sin n
(c)
(d)
(e)
(f)
(2) Using the integral test show that
1
n=2 n log n
diverges.
Rought Solutions
Homework 5: Due Oct 25th (Thursday)
(1) Show that the following series convergent. How many terms of the series
do we need to add in order to nd the sum to the indicated accuracy?
(1)n
(a)
( error less than 0.00005)
n=1 n6
(1)n
(b)
n=1 n5n ( error less
Homework 3: Due Oct 2nd
(1) Rewrite the following series using the Sigma notation.
1
1
(a) 3 + 1 + 27 +
9
1
1
1
1
(b) 1 + 4 + 9 + 16 + 25 +
(1)n
(2) It is known that n=1 n converges. Find
(1)n
(1)n
.
n
n
n=1
n=3
(3) Prove that the following series dive