Finding The Limit As N Approaches Infinity Of (1+1/N)ⁿ: Deriving Euler’S Number.

the limit as n approaches infinity of (1+1/n)ⁿ is

the limit as n approaches infinity of (1+1/n)ⁿ is equal to e.

The limit as n approaches infinity of (1+1/n)ⁿ is equal to the mathematical constant e, which is approximately equal to 2.71828. This limit is often referred to as the Euler’s number and is widely used in calculus and mathematical analysis.

To derive this result, we can use the following steps:

Step 1: We can rewrite the expression (1+1/n)ⁿ as (1/n + 1)ⁿ.

Step 2: We can apply the binomial theorem to the expression (1/n + 1)ⁿ to get:

(1/n + 1)ⁿ = ∑(n choose k) (1/n)ⁿ-k (1)ⁿ for k = 0 to n,

where (n choose k) is the binomial coefficient.

Step 3: As n approaches infinity, we can see that the term (1/n)ⁿ-k approaches zero for all values of k other than k=0. Thus, we can simplify the above expression to:

(1/n + 1)ⁿ = 1 + (1/n)ⁿ * ∑(n choose k) (1/n)⁻ᵏ (1)ⁿ for k = 1 to n.

Step 4: As n approaches infinity, we can see that the sum in the above expression approaches the exponential function e, defined as:

e = ∑(k=0 to infinity) (1/k!) = 1 + 1/1! + 1/2! + 1/3! + …

Thus, we can simplify the above expression as:

(1/n + 1)ⁿ = 1 + (1/n)ⁿ * e.

Step 5: Taking the limit as n approaches infinity, we can see that the term (1/n)ⁿ approaches zero. Thus, we get:

lim (1+1/n)ⁿ = lim (1/n + 1)ⁿ = lim (1 + (1/n)ⁿ * e) = 1 + 0 * e = e.

Therefore, the limit as n approaches infinity of (1+1/n)ⁿ is equal to e.

More Answers: