Consider the series ∑n=1∞5n2+n.
2 answers:
(a) Decompose the summand into partial fractions:




The <em>n</em>-th partial sum for the series is

which can be simplified significantly by examinging consective terms in the sum:


(b) Using the result of (a), you then get

(c) As shown in (a), the partial sum is simplified because of the reasons given in options A and D, and the result of (b) says that B is also correct.
Answer:
Part a. 
Part b. 
Part c. A, B, and D
General Formulas and Concepts:
<u>Algebra I</u>
- Terms/Coefficients
- Factoring
<u>Pre-Calculus</u>
- Partial Fraction Decomposition
<u>Calculus</u>
Limits
- Limit Rule [Variable Direct Substitution]:

- Limit Property [Addition/Subtraction]:
![\displaystyle \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20c%7D%20%5Bf%28x%29%20%5Cpm%20g%28x%29%5D%20%3D%20%20%5Clim_%7Bx%20%5Cto%20c%7D%20f%28x%29%20%5Cpm%20%5Clim_%7Bx%20%5Cto%20c%7D%20g%28x%29)
Sequences
Series
- Definition of a convergent or divergent series
Telescoping Series: 
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>

<u>Step 2: Rewrite Sum</u>
- Factor:

- Break up [Partial Fraction Decomposition]:

- Simplify [Common Denominator]:

- [Decomp] Substitute in <em>n</em> = 0:

- Simplify:

- [Decomp] Substitute in <em>n</em> = -1:

- Simplify:

- Solve:

- [Decomp] Substitute in variables:

- Simplify:

- Substitute in decomp [Sum]:

<u>Step 3: Find Sum</u>
- Find Sₙ terms:

- Find general Sₙ formula:

- Find Sum [Take limit]:

- Evaluate limit [Limit Rule - Variable Direct Substitution]:

- Simplify:

∴ the sum converges by the Telescoping Series.
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Convergence Tests (BC Only)
Book: College Calculus 10e
You might be interested in
This is a combination question
Hope this helps
Answer:
Restudy and try to take it again.
Step-by-step explanation:
Try watching math videos and do extra studying on the areas you missed on your last attempt.
Answer:
178
Step-by-step explanation:
Step-by-step explanation:
<h2>
<em><u>Thank</u></em><em><u> you</u></em></h2><h2>
<em><u>t</u></em><em><u>his </u></em><em><u>is </u></em><em><u>your</u></em><em><u> </u></em><em><u>answer.</u></em></h2><h2>
<em><u>mark </u></em><em><u>Brainlist</u></em><em><u> </u></em><em><u>please</u></em></h2>
<em>Greetings from Brasil...</em>
Be that unknown number X
Then we can write the following expression:
X + 4.7 = 8.1
<h2>X = 3.4</h2>