Let Sn = bk. Use the definition of s6 and your work in part (b) to write out an unsimplified expression 2 2 2 7 8 e) i for s6. Then show how to simplify this sum to obtain s6 = 2 + 2 k=1 f) Look for the pattern in your work in part (e) and then use the pattern to write an explicit formula for sn. Make sure you show how you obtain this explicit formula from the definition of sn. Determine a function f(x) such that f(n) = Sn when n is a natural number.
Let Sn = bk. Use the definition of s6 and your work in part (b) to write out an unsimplified expression 2 2 2 7 8 e) i for s6. Then show how to simplify this sum to obtain s6 = 2 + 2 k=1 f) Look for the pattern in your work in part (e) and then use the pattern to write an explicit formula for sn. Make sure you show how you obtain this explicit formula from the definition of sn. Determine a function f(x) such that f(n) = Sn when n is a natural number.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please show work thank you! Will also submit one more section of this after first two is completed!
Transcribed Image Text:e) I
Let Sn=
bk. Use the definition of s6 and your work in part (b) to write out an unsimplified expression
k=1
for s6. Then show how to simplify this sum to obtain s6 = 2 +
2 2
NIN
2
VIN
2100
8
f)
Look for the pattern in your work in part (e) and then use the pattern to write an explicit formula for sn.
Make sure you show how you obtain this explicit formula from the definition of sn. Determine a function f(x) such that
f(n) = sn when n is a natural number.
Transcribed Image Text:7) Complete the following parts to explore and better understand
• The sequence bn
●
a)
b)
The sequence Sn = Σb, of partial sums
k=1
d)
n
limb, the limit of the sequence of partial sums
k=
n→∞
b₁ =
b₂ =
b3
c) 1
=
lim bn =
n→∞
Let bn
4
n² + 2n
Use partial fractions to show that bn
=
Write out the first 6 terms of bn
2
n
2
n+2
b4 =
Do not simplify.
b5 =
2
b6
n
2
n+2
Show all intermediate steps.
Modify this Desmos Interactive to graph the sequence bn. Based on your graph, estimate lim bn.
n→∞
Determine lim bn directly using the formula for bn. Write a sentence to explain your reasoning.
n→∞
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Follow-up Question
Transcribed Image Text:Determine lim bn directly using the formula for bn. Write a sentence to explain your reasoning.
n→∞
Solution
by Bartleby ExpertFollow-up Question
Please show all work, thanks!
![7) Complete the following parts to explore and better understand
- The sequence \( b_n \)
- The sequence \( s_n = \sum_{k=1}^{n} b_k \) of partial sums
- \( \lim_{n \to \infty} \sum_{k=1}^{n} b_k \), the limit of the sequence of partial sums
a) Let \( b_n = \frac{4}{n^2 + 2n} \). Use partial fractions to show that \( b_n = \frac{2}{n} - \frac{2}{n+2} \). Show all intermediate steps.
b) Write out the first 6 terms of \( b_n = \frac{2}{n} - \frac{2}{n+2} \). Do not simplify.
- \( b_1 = \)
- \( b_2 = \)
- \( b_3 = \)
- \( b_4 = \)
- \( b_5 = \)
- \( b_6 = \)
c) Modify this Desmos Interactive to graph the sequence \( b_n \). Based on your graph, estimate \( \lim_{n \to \infty} b_n \).
\[ \lim_{n \to \infty} b_n = \]
d) Determine \( \lim_{n \to \infty} b_n \) directly using the formula for \( b_n \). Write a sentence to explain your reasoning.](https://content.bartleby.com/qna-images/question/35822848-d2b2-46f4-98be-d0852bf0c323/aa17df79-37a3-413f-991b-249ffdd41611/u3nvvm8_processed.png)
Transcribed Image Text:7) Complete the following parts to explore and better understand
- The sequence \( b_n \)
- The sequence \( s_n = \sum_{k=1}^{n} b_k \) of partial sums
- \( \lim_{n \to \infty} \sum_{k=1}^{n} b_k \), the limit of the sequence of partial sums
a) Let \( b_n = \frac{4}{n^2 + 2n} \). Use partial fractions to show that \( b_n = \frac{2}{n} - \frac{2}{n+2} \). Show all intermediate steps.
b) Write out the first 6 terms of \( b_n = \frac{2}{n} - \frac{2}{n+2} \). Do not simplify.
- \( b_1 = \)
- \( b_2 = \)
- \( b_3 = \)
- \( b_4 = \)
- \( b_5 = \)
- \( b_6 = \)
c) Modify this Desmos Interactive to graph the sequence \( b_n \). Based on your graph, estimate \( \lim_{n \to \infty} b_n \).
\[ \lim_{n \to \infty} b_n = \]
d) Determine \( \lim_{n \to \infty} b_n \) directly using the formula for \( b_n \). Write a sentence to explain your reasoning.
Solution
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