i just through about 2^n being 2×2×2×... } n times and n! being (n)(n-1)(n-2)(n-3)..(2)(1), where n! grows the multipliers faster than 2^n, so it tends to 0 as the divider gets bigger
The way I did it (probably not very correct but still): Σ[n=0,∞] 2^n/n! = e² (by the Maclaurin series for e^x), or in other words, the series converges. With that, the limit of the inside function must be 0 for it to bypass the test for divergence.
You just provided a way to prove that an exponential with any base is negligible against the factorial. In other words, we can replace the 3 with any other number, and the limit is still 0. Nice!
2^n / n! At n=1, we get 2/1 At n=2, we get 4/2 At n=3, we get 8/6 At n=4, we get 16/24 At n=5, we get 32/120 At n=6, we get 64/720 At n=7, we get 128/5040 We can make the observation that it's clear that the higher 'N' gets, the smaller this ratio becomes. Hence, it is getting closer to 0 as 'N' gets higher, so Lim 2^n/n! = 0 n => INF
i just through about 2^n being 2×2×2×... } n times and n! being (n)(n-1)(n-2)(n-3)..(2)(1), where n! grows the multipliers faster than 2^n, so it tends to 0 as the divider gets bigger
The way I did it (probably not very correct but still):
Σ[n=0,∞] 2^n/n! = e² (by the Maclaurin series for e^x), or in other words, the series converges.
With that, the limit of the inside function must be 0 for it to bypass the test for divergence.
Very nice!
You just provided a way to prove that an exponential with any base is negligible against the factorial. In other words, we can replace the 3 with any other number, and the limit is still 0.
Nice!
Stirling
2^n / n!
At n=1, we get 2/1
At n=2, we get 4/2
At n=3, we get 8/6
At n=4, we get 16/24
At n=5, we get 32/120
At n=6, we get 64/720
At n=7, we get 128/5040
We can make the observation that it's clear that the higher 'N' gets, the smaller this ratio becomes.
Hence, it is getting closer to 0 as 'N' gets higher, so
Lim 2^n/n! = 0
n => INF
That's false reasoning. What you saw is that the value of the fractions are decreasing. You do not know that they are approaching zero from your list.
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