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 'Significant figures for inherently bound values'

I have been insisting to my statistics students that for probabilities, the rule is the number of significant figures is the number of digits past the leading zeros or leading nines. For example to give 4 significant figures for a probability: 0.000001234 and 0.99999991234 are the correct number of decimal places. That way the complementary probability can also be given to the same significant figures ( 0.999998766 and 0.00000008766 respectively). More generally if you have a value that...
View full post »

Similar threads

A 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
I 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
POTW Power Sums Limits: Evaluating $\lim_{n\to \infty}$
  • · Replies 3 ·
Replies
3
Views
2K
B Does undefined always mean infinity?
  • · Replies 15 ·
Replies
15
Views
4K
I Proving the limit of a sequence from the definition of limit
  • · Replies 16 ·
Replies
16
Views
2K