MAT272 Test3A SP19 KEY.

Math 272 Test 3A Spring 2019 Name: KEY

Clearly indicate your method of solution for full credit. Clearly indicate your final answers. Time limit is 100 minutes.
Non-graphing calculators allowed. (8 points)
1. Complete the following table, using DNE (does not exist) as necessary.
Sequence Increasing, Decreasing or Bounds lim 𝑎𝑛
𝑛→∞
Neither?
 1
n 1 Neither Upper: 0
an  n
, n 1 1/2 (or larger)
2


Lower:
-1/4 (or smaller)



n Increasing Upper: 1
an  , n 1 1 (or larger)
n3



Lower:
1/4 (or smaller)




2. Find the explicit formula for the 𝑛𝑡ℎ term of the sequence below.
2 4 8 16 32 
 ,  , ,  , ... (4 points)
 7 12 17 22 27 



(−𝟏)𝒏+𝟏 𝟐𝒏
𝒂𝒏 =
𝟐 + 𝟓𝒏




This study source was downloaded by 100000845196002 from CourseHero.com on 05-26-2022 14:00:48 GMT -05:00


https://www.coursehero.com/file/40300117/MAT272-Test3A-SP19-KEYpdf/

, 3. Use a convergence test of your choice to determine whether the following series converges absolutely, converges
conditionally or diverges. State the test you used and show all work. (7 points)
 1
k


k 2 ln k


Test for absolute convergence:
(−𝟏)𝒌 𝟏 𝟏
Comparison Test: |
𝒍𝒏 𝒌
| = 𝒍𝒏 𝒌 > 𝒌 (diverges)

The series does not converge absolutely.

Test for conditional convergence:

Alternating Series Test:
𝟏 𝟏 𝟏
1) 𝐥𝐧 𝒌
is decreasing since 𝐥𝐧(𝒌+𝟏) < 𝐥𝐧 𝒌
𝟏
2) 𝐥𝐢𝐦 𝐥𝐧 𝒌 = 𝟎
𝒌→∞

The series does converge conditionally.



Therefore the series converges conditionally (but not absolutely).



4. Calculate and write the partial sum of the first four non-zero terms of the Taylor Series representation of
f ( x)  e3 x centered at a  2 . (8 points)



𝒏 𝒇(𝒏) (𝒙) 𝒇(𝒏) (𝟐)
0 𝒆𝟑𝒙 𝒆𝟔
1 𝟑𝒆𝟑𝒙 𝟑𝒆𝟔
2 𝟗𝒆𝟑𝒙 𝟗𝒆𝟔
3 𝟐𝟕𝒆𝟑𝒙 𝟐𝟕𝒆𝟔


𝒆𝟔 𝟑𝒆𝟔 𝟗𝒆𝟔 𝟐𝟕𝒆𝟔
𝑻𝟑 (𝒙) = (𝒙 − 𝟐)𝟎 + (𝒙 − 𝟐)𝟏 + (𝒙 − 𝟐)𝟐 + (𝒙 − 𝟐)𝟑
𝟎! 𝟏! 𝟐! 𝟑!
𝟗 𝟗
𝑻𝟑 (𝒙) = 𝒆𝟔 + 𝟑𝒆𝟔 (𝒙 − 𝟐) + 𝟐 𝒆𝟔 (𝒙 − 𝟐)𝟐 + 𝟐 𝒆𝟔 (𝒙 − 𝟐)𝟑




This study source was downloaded by 100000845196002 from CourseHero.com on 05-26-2022 14:00:48 GMT -05:00


https://www.coursehero.com/file/40300117/MAT272-Test3A-SP19-KEYpdf/