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Calculus Level 2

lim ⁡ n → ∞ ( 1 + 1 n ) n \large \lim_{n\to\infty} \left(1+\frac1n\right)^n

Evalute the limit above to 3 decimal places.


The answer is 2.718.

Syed Baqir by Syed Baqir , 24, Cyprus

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3 solutions

Raj Rajput
Jul 16, 2015

3 Helpful 0 Interesting 0 Brilliant 0 Confused

The value of the limit comes out to be e = 2.718... for reasons described here

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Yes, it's easier to relate the limit to the polynomial expansion of e e .

Yep. Exponential growth slows down. 

1
2
 
>>> for n in xrange(0,40):
        print (1 + 1 / float(2 ** n)) ** (2 ** n)


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2.0
2.25
2.44140625
2.56578451395
2.63792849737
2.67699012938
2.69734495257
2.70773901969
2.71299162425
2.71563200017
2.71695572947
2.71761848234
2.71795008119
2.71811593627
2.71819887772
2.71824035193
2.7182610899
2.71827145911
2.71827664377
2.71827923611
2.71828053228
2.71828118037
2.71828150441
2.71828166644
2.71828174745
2.71828178795
2.71828180821
2.71828181833
2.7182818234
2.71828182593
2.71828182719
2.71828182783
2.71828182814
2.7182818283
2.71828182838
2.71828182842
2.71828182844
2.71828182845
2.71828182845
2.71828182846

Arulx Z - 5 years, 11 months ago
Krishna Shankar
Jul 1, 2015

The value of the limit comes out to be e = 2.718

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

Why?

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