sequence series project fall 2013 ;calculus 2 math 2414
julian kay
sequence_series_project_fall_2013.pdf
Name ____________________________ Calculus 2 Mike Huff
Tools: Calculator, scrap paper, textbook, class notes, other students in the class, and the professor. No other sources please! You may check your work using calculators or computers but you must show each step of the calculation for full credit. You may share ideas but you may not copy another student’s work. You may get help from the lab on the techniques but they may not work out any of these problems for you.
Deadline: _________________________
Calculus 2 Math 2414 Project 2 Fall 2013 Mike Huff
Do all work on paper for partial credit. Good luck!!
(8 points)
1. Determine whether each sequence converges or diverges, giving a reason,
and if it does converge, find its limit.
a)
2 na n n n= − −
b)
ln( 1) n
n a
n +
=
c) ( )1 n
n n
a n −
= d) 1
1 1n na dxn x
= ∫
(4 points)
2. Find the sum of the convergent series: ( )( )1
6 2 1 2 1n n n
∞
= − + ∑
(4 points)
3. Use the Integral Test to determine the convergence or divergence of the
series: 2 1 1
n
n n
e e
∞
= + ∑
(9 points)
4. Determine whether each series is convergent or divergent. If it is convergent, find its sum.
a) 1
2 n
n
n n
∞
=
−⎛ ⎞ ⎜ ⎟ ⎝ ⎠
∑
b) 1
1 1 2 3n n
∞
= + + + + ∑
c) 1
0
2 3 5 4
n n
n n n
+∞
=
⎛ ⎞ +⎜ ⎟
⎝ ⎠ ∑
(6 points)
5. Determine whether each series converges absolutely, converges
conditionally, or diverges. Carefully explain your reasoning for full credit.
a) 1
4 1
( 1)n
n n
+∞
=
− ∑
b) 1 1
ln ( 1)n
n
n n
∞ +
=
−∑
(8 points)
6. a) Find a geometric power series representation for the function 3
( ) 2
f x x
= +
and find its radius of convergence.
b) Express 2 1 1
ln(1 ) 1 1
x dx dx x x
− = − + −∫ ∫ as a power series using know
power series.
(6 points)
7. a) Show that the series converges: 1
1
( 1) 2 1
n
n n
+∞
=
− +∑
b) Find the 25th partial sum: 25s .
c) Find the error bound in your approximation.
(4 points)
8. There is absolutely no empirical evidence for the divergence of the harmonic series even though we know that it diverges. The partial sums,
which satisfy the inequality 1
1 1
1 1 1 1 ln( 1) 1 1 1 ln( )
2
n n
n dx dx n x n x
+
+ ≤ ≤ + + + ≤ + = +∫ ∫ ,
just grow too slowly. To see what I mean, suppose you had started with
1 1s = the day the universe was formed, thirteen billion years ago, and
added a new term every second. About how large would ns be today?
(6 points)
9. Find the radius and interval of convergence of the power series ( )
1
4 3
n
n n
x n
∞
=
+ ∑ .
Radius of convergence ________________
Interval of convergence ________________
Converges absolutely on the interval ________________
Converges conditionally when x is ________________
(9 points)
10. Power Series from Known Series a. Calculate the first two non-vanishing terms in the Taylor series for sinx
about = 0x .
b. Use the result of a) to obtain an approximate to ( )= ∫ 2 0
( ) sin x
F x t dt that is
valid for small x.
b. Express − 3xxe as a Taylor series. What is its radius of convergence?
(8 points)
11. Find the exact sum of each series.
a. 0
2 3 !
n
n n n
∞
= ∑
b. ( ) ( )
2 1
2 1 0
1
3 2 1 !
n n
n n n
π +∞ +
=
−
+ ∑
c. ( ) ( )
2
2 0
1
4 2 !
n n
n n n
π∞
=
− ∑
d. ( ) 0
1 !
n n
n
e n
∞
=
− ∑
(6 points)
12. Find the Taylor polynomial of degree 2 at =x e for the function
defined by = 2( ) lnf x x x .
(6 points)
13. Find the 4th -degree Taylor polynomial centered at = 0x for the
function = +( ) 1f x x . Use the polynomial to approximate 1.25 .
Compare the value to the value obtained by your calculator.
14. (6 points)
Use a power series to approximate 21
0 xe dx−∫ with an error of less than
0.01.
15. (6 points)
Use a power series to approximate 6
0.5
0
1 1 x
dx +∫ with an error of less than
0.0001.
(6 points)
16. Find the first three terms of the Taylor series for ( ) cscf x x= centered
at 2
a π
= . Please show all work.
