Math 55 HW 4 Sequences and Series

Mathematics 55

1st Semester, Semester, A.Y. 2016-2017 2016-2017

Homework 4

Work independently independently.. Do not consult consult anyone anyone except your instructor. instructor. Provide neat and clear solutions. Due: Due: 18 Nove November mber 2016 (Frid (Friday ay), ),   2:45 PM. Late Late subm submis issi sion onss will will merit merit deducti deductions ons.. Use Use yelyellow pad for your work. Do as indicated. 1. Let

an

=e

n

−

1

− tan (−n). −

+∞

(a) Determine whether the sequence

n n=1  is

 {a }

convergent.

(2 points)

+∞

(b) Determine whether the series



an  is

convergent.

(1 point)

n=1

+∞

2. Conside Considerr the series series

−

( 1)n

n=1

3n √  . 3n n + 1 3

(3 points)

(a) Show that the given series is convergent.

(b) Using the Limit Comparison Test, Test, show that the series is conditionally conditionally convergent. convergent. points)

(3 

3. Determine Determine whether the following following series are convergent convergent or divergent. divergent. Use necessary tests. tests. Make sure to verify conditions needed in using certain tests. +∞

(a)

e1/n

  − ·· − n=1 +∞

(b)

n=1 +∞

(c)

n=1 +∞

(d)

using the Integral Integral Test Test

(6 points)

5n  cos2 n

(3 points)

n2

2n

2  5  8  11 (3n n(n!)

· ···

− 1)

(4 points)

3

( 1)n 1 (en

n=1

−

(4 points)

n3n

BONUS. +∞

1. Consider Consider the telescoping series series

 n=1

(a) Find the

k th

5 2n

− 1 −



5 . 2n + 1

(4 points)

partial sum of the series.

(b) Use your answer in (a) to find the sum of the series.

      +∞

2. Is the series

n=1

π e

n

+

1

n

7/5

 convergent or divergent? Justify.

(3 points)