Determine the convergence or divergence of the series. 2 n(n+3) Step 1 Decompose to partial fractions. Step 2 n=1 n=1 Write the sum Sp. Step 4 n(n+3) Therefore, Sn = [(-¹)+(-) + (− ² ) + ( ² - Sn= Σ( n=1 1 = ²/3 [ ( ² + + + + + + / -¯- 0 + ¹ ) + (n ² ₂ - 0 ²+ ₁) + (n ² ₁ - ² + ₂) + (1 - 1 Evaluate the sum. n=1 2 (n)(n+3) Step 3 Observe that appears as both positive and negative in the series. Thus, it will get cancelled. Similarly, will also get cancelled. The terms like 1, 2, and that do not have their additive inverses will not get cancelled. 7. 3 = lim Sn n-* lim n10 n+3 n+3 n+3 X n+3 ²3 ( + + + + + + + ² - ² + ₁ - X n+2 7 + ... n+3 n+3)]. -n+3)
Determine the convergence or divergence of the series. 2 n(n+3) Step 1 Decompose to partial fractions. Step 2 n=1 n=1 Write the sum Sp. Step 4 n(n+3) Therefore, Sn = [(-¹)+(-) + (− ² ) + ( ² - Sn= Σ( n=1 1 = ²/3 [ ( ² + + + + + + / -¯- 0 + ¹ ) + (n ² ₂ - 0 ²+ ₁) + (n ² ₁ - ² + ₂) + (1 - 1 Evaluate the sum. n=1 2 (n)(n+3) Step 3 Observe that appears as both positive and negative in the series. Thus, it will get cancelled. Similarly, will also get cancelled. The terms like 1, 2, and that do not have their additive inverses will not get cancelled. 7. 3 = lim Sn n-* lim n10 n+3 n+3 n+3 X n+3 ²3 ( + + + + + + + ² - ² + ₁ - X n+2 7 + ... n+3 n+3)]. -n+3)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
Related questions
Question
![Determine the convergence or divergence of the series.
Step 1
Step 2
Decompose to partial fractions.
Step 3
∞
n = 1
n = 1
Step 4
Write the sum Sn.
Therefore,
2
n(n + 3)
Observe that
2
n(n + 3)
Sn = ²3² [( ²¹ ) + (-1/2-1/2) + (-1)+(²+3x
4
n = 1
1
1
1
1
+ (n ² ₂ - 0 ²¹² ₁) + (n ² ₁ - ² / ₂ ) + (-1/2 -
+ 1
²
1
- 2
n+ 2
Sn = ²3/[(1/2
+
Evaluate the sum.
Σ(-
n = 1
21/12/23
2
(n)(n+3)
+
appears as both positive and negative in the series. Thus, it will get cancelled. Similarly,... will also get cancelled. The terms like 1, and
2'
1
3
=
lim Sn
1
n + 1
n→∞
= lim
2/1
n→∞ 37
W|N
3
1
11
2
n+3
n+3
1
1
1
3 n+ 1
n+3
n+2
1
n+ 3
1
n+3)].
n+ 3
that do not have their additive inverses will not get cancelled.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fdef185-de8b-42f5-933e-8018dcf36541%2F2d9cc296-f751-47e1-9141-b49ade2525fb%2F9nvyyl_processed.png&w=3840&q=75)
Transcribed Image Text:Determine the convergence or divergence of the series.
Step 1
Step 2
Decompose to partial fractions.
Step 3
∞
n = 1
n = 1
Step 4
Write the sum Sn.
Therefore,
2
n(n + 3)
Observe that
2
n(n + 3)
Sn = ²3² [( ²¹ ) + (-1/2-1/2) + (-1)+(²+3x
4
n = 1
1
1
1
1
+ (n ² ₂ - 0 ²¹² ₁) + (n ² ₁ - ² / ₂ ) + (-1/2 -
+ 1
²
1
- 2
n+ 2
Sn = ²3/[(1/2
+
Evaluate the sum.
Σ(-
n = 1
21/12/23
2
(n)(n+3)
+
appears as both positive and negative in the series. Thus, it will get cancelled. Similarly,... will also get cancelled. The terms like 1, and
2'
1
3
=
lim Sn
1
n + 1
n→∞
= lim
2/1
n→∞ 37
W|N
3
1
11
2
n+3
n+3
1
1
1
3 n+ 1
n+3
n+2
1
n+ 3
1
n+3)].
n+ 3
that do not have their additive inverses will not get cancelled.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning