The binomial series coefficient

Click For Summary
The discussion focuses on using the binomial series to expand the function 1/(2+x)^3 as a power series. A participant seeks clarification on the reasoning behind the transition from the second to the third line of their work, particularly regarding the role of the (-1)^n term and the introduction of (n+1). Another participant confirms that the (n+1) term is consistently present and explains its visibility in the series expansion. They also suggest including the "n" term for clarity on cancellations with n!. The conversation emphasizes understanding the structure of the binomial series expansion for accurate calculations.
Aerospace93
Messages
22
Reaction score
0

Homework Statement


Use the binomial series to expand the function as a power series.
1/(2+x)3

I have attached an image. I understand until the end of the second line. I don't see the reasoning used to follow through to the third line. the (-1)^n is because the sign alternates becoming negative for every odd term. Are the 2's put on the top and the bottom so then the top 2, along with the number preceding it cancel with the (n!)? I have no clue how the (n+1) comes into play.

I would appreciate all the help!
 

Attachments

  • binomial coefficient.png
    binomial coefficient.png
    16.5 KB · Views: 601
Physics news on Phys.org
Yes, that is exactly the reason. The (n + 1) is there all the time, it is the number before (n + 2) :-) Only in the second line it is still absorbed in the ... dots, while in the second line it is included explicitly. I would also have included the "n" term so it is completely obvious which part cancels out against n!.
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
View full post »

Similar threads

How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
A few questions to make sure I understand Fourier series
  • · Replies 3 ·
Replies
3
Views
2K
Multiplication of Taylor and Laurent series
  • · Replies 12 ·
Replies
12
Views
3K
Maclaurin Series When 0 is in the Denominator?
  • · Replies 48 ·
2
Replies
48
Views
5K
Why the series is divergent based on the Preliminary test
  • · Replies 2 ·
Replies
2
Views
1K
On pointwise convergence of Fourier series
  • · Replies 3 ·
Replies
3
Views
2K
Binomial Distribution Question
  • · Replies 5 ·
Replies
5
Views
2K
Inverse Binomial Expansion within Laurent Series?
  • · Replies 2 ·
Replies
2
Views
2K
Variance of binomial distribution
  • · Replies 8 ·
Replies
8
Views
2K
Finding a periodic solution to ODE
  • · Replies 16 ·
Replies
16
Views
2K