Understanding the (1+1/n)^n Problem: The Last Term in Binomial Expansion

  • Thread starter MathematicalPhysicist
  • Start date
In summary, the conversation discussed the difference between the last term in the binomial expansion of (1+1/n)^n according to the text and the actual correct form. The first form provided by the text is (1/n!)(1-1/n)(1-2/n)...(1-(n-1)/n), while the correct form is (1/n!)(n(n-1)...1)/n^n. The speaker pointed out that the two forms are almost identical, with the only difference being the first form is multiplied by n. The problem was quickly resolved and it was concluded that the two forms are indeed the same.
  • #1
MathematicalPhysicist
Gold Member
4,699
373
it's rather a bypass question, not a direct question about 'e'.

the last term in the binomial expansion of (1+1/n)^n is according to my text:
(1/n!)(1-1/n)(1-2/n)...(1-(n-1)/n)
while it should be (1/n!)(n(n-1)...1)/n^n

okay, these two terms are almost identical, the first equals
(with disregarding of 1/n!):
((n-1)(n-2)...1)/n^n
and second one is the first multiplied by n, then what is the problem.

i assume it's with me ( just kidding).
 
Physics news on Phys.org
  • #2
(1/n!)(1-1/n)(1-2/n)...(1-(n-1)/n)
= (1/n!)(n-1)(n-2)...(n-(n-1))/n^(n-1)
= (1/n!)(n-1)...1/n^(n-1)
= (1/n!)(n(n-1)...1)/n^n

so, they're the same
 
  • #3
okay i think i can see it, thanks.
 

FAQ: Understanding the (1+1/n)^n Problem: The Last Term in Binomial Expansion

What is the (1+1/n)^n problem in binomial expansion?

The (1+1/n)^n problem in binomial expansion is a mathematical concept that arises when expanding the expression (1+1/n)^n using the binomial theorem. The problem is trying to determine the value of the last term in the expansion as n approaches infinity.

Why is the (1+1/n)^n problem important?

The (1+1/n)^n problem has significant applications in fields such as statistics, economics, and physics. It is also a fundamental concept in calculus and helps in understanding the behavior of functions as their inputs approach infinity.

How is the (1+1/n)^n problem solved?

The (1+1/n)^n problem can be solved using various methods, such as using the binomial theorem or using limits. The exact value of the last term in the expansion can be found by taking the limit as n approaches infinity. However, in most cases, an approximation is used due to the complexity of the calculation.

What is the significance of the last term in the (1+1/n)^n expansion?

The last term in the (1+1/n)^n expansion is significant because it represents the behavior of the function as n approaches infinity. It also plays a crucial role in determining the convergence or divergence of the series and in approximating the value of the function at large inputs.

How does the (1+1/n)^n problem relate to other mathematical concepts?

The (1+1/n)^n problem is closely related to other mathematical concepts such as limits, series, and calculus. It also has connections to real-world applications, such as compound interest and population growth, making it a valuable concept in various fields of study.

Similar threads

Coefficients in the expansion of (1+x)n in A.P, find n
Replies
6
Views
1K
N-dimensional broken stick problem -- find joint density
Replies
3
Views
681
Proving that ##\lim [\sqrt{4n^2 +n} - 2n] = \frac{1}{4}##
Replies
9
Views
2K
Solving the Density of States: Understanding dn/dE
Replies
7
Views
1K
Find the ##r^{th}## term of ##(a+2x)^n##
Replies
3
Views
1K
Why smooth spherical waves with attenuation are only possible in 3-D
Replies
1
Views
1K
Simplification step: solving a diff eqn using a power series
Replies
9
Views
1K
Getting the coefficients of inhomogeneous PDE using Fourier method
Replies
6
Views
2K
Optimization: How to find the dual problem?
Replies
6
Views
2K
Find the sum of the coefficients in the expansion ##(1+x)^n##
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top