MHB Limit of a Geometric Sequence: How to Evaluate?

  • Thread starter Thread starter tmt1
  • Start date Start date
  • Tags Tags
    Limit
Click For Summary
The limit of the geometric sequence is evaluated as follows: the limit of (2/3)^n as n approaches infinity is indeed 0. This is supported by the fact that the exponential function e^n grows faster than any polynomial or geometric sequence with a base less than 1. The transformation of (2/3)^n into an exponential form using natural logarithms confirms that it approaches zero. Thus, the limit can be concluded without extensive calculations. The final result is that the limit is 0.
tmt1
Messages
230
Reaction score
0
I have this limit:

$$\lim_{{n}\to{\infty}} {(\frac{2}{3})}^{n}$$

I know the answer is 0 but how can I evaluate this?
 
Mathematics news on Phys.org
Can you agree that $\lim_{n\to\infty}\dfrac{1}{e^n}=0$, without calculation?

If so,

$$\lim_{n\to\infty}\left(\dfrac23\right)^n=\lim_{n\to\infty}e^{n\ln(2/3)}=\lim_{n\to\infty}e^{-n\ln(3/2)}$$

$$=\lim_{n\to\infty}\left(\dfrac{1}{e^n}\right)^{\ln(3/2)}=0^{\ln(3/2)}=0$$
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

Graduate Limit of ##i^\frac{1}{n}## as ##n \to \infty##
  • · Replies 14 ·
Replies
14
Views
2K
MHB How to fully solve this limit evaluation using integration?
  • · Replies 7 ·
Replies
7
Views
2K
MHB What is the solution to the exponential series limit problem?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Evaluate Limit: $$\lim_{x\to\infty} (-1)^nn^3 + 2^{-n}$$
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Evaluation of the sum 1^m+3^m+5^m+ ....................(2n+1)^{m}
  • · Replies 1 ·
Replies
1
Views
2K
MHB Evaluate Product: Limit of Product Involving Tangents
  • · Replies 2 ·
Replies
2
Views
1K
MHB Limit Product Evaluation: $\displaystyle \infty$
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Power Sums Limits: Evaluating $\lim_{n\to \infty}$
  • · Replies 3 ·
Replies
3
Views
2K
High School Does undefined always mean infinity?
  • · Replies 15 ·
Replies
15
Views
4K
Undergrad Proving the limit of a sequence from the definition of limit
  • · Replies 16 ·
Replies
16
Views
2K