Hello all!
Consider Lim n tends to infinity (1+1/n)^n .my guess is as n tends to infinity the eqn becomes (1+1/infinity)^infinity =1.but the right answer is e(I am aware of the mathematical proofs) but how different is that approach from mine?

 

You identified the answer in your question with the word "guess". In mathematics we only use a "guess" to lead us to an answer. We still have to find the proof. Where your intuition fails you is the rate at which 1/n gets small versus the rate at which (1+1/n) approaches 1

 

Hello all!
Consider Lim n tends to infinity (1+1/n)^n .my guess is as n tends to infinity the eqn becomes (1+1/infinity)^infinity =1.but the right answer is e(I am aware of the mathematical proofs) but how different is that approach from mine?

Start with n=1 then calculate the result.
Then n=2 and calculate the result.
Then n=3 and calculate the result.
You can then go by 10's: 10, 20, 30, etc.
Look at how the results increase then start to level off around 2.7 and that will give you some insight.

 

Thanks a lot to both of you .

 

Thanks a lot to both of you .

May be you are approaching the problem as follows: (1+1/infinity)^infinity which is equal to (1+0)^infinity which is turn is 1^infinity which is 1...!!!

However, the 1^infinity is NOT 1...!!! Its indeterminate...!!

 

https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function

 

May be you are approaching the problem as follows: (1+1/infinity)^infinity which is equal to (1+0)^infinity which is turn is 1^infinity which is 1...!!!

However, the 1^infinity is NOT 1...!!! Its indeterminate...!!

This reminds me of an old math joke. One way to demonstrate that the answer is indeterminate, is to note that

1^∞ = 0​

for very small values of 1 !!!

similar to 2+2=5, for large values of 2 and small values of 5.

 

May be you are approaching the problem as follows: (1+1/infinity)^infinity which is equal to (1+0)^infinity which is turn is 1^infinity which is 1...!!!

However, the 1^infinity is NOT 1...!!! Its indeterminate...!!

Thank u very much!!!

 

(1+1/100)^100 = 2.704813829422
(1+1/1000)^1000 = 2.716923932236
(1+1/10000)^10000 = 2.718145926825
(1+1/100000)^100000 = 2.718268237174
(1+1/1000000)^1000000 = 2.718280469319
(1+1/10000000)^10000000 = 2.718281692544

 

https://keisan.casio.com/calculator : (1-1/1E+23)^(1E+23) = 0.3678794411714423215955219307642550102342031534
calc.exe (Windows Calculator) : Invalid input _ _ _ _ exp(-1) = 0.36787944117144232159552377016146