Problem 98 - Essential Calculus

Question

Compute the first six terms of the sequence \(\left\\{a_{n}\right\\}=\left\\{\left(1+\frac{1}{n}\right)^{n}\right\\}\) If the sequence converges, find its limit.

Step-by-Step Solution

Answer

The first six terms of the sequence are approximately 2, 2.25, 2.37, 2.44, 2.49 and 2.52. The sequence appears to converge, and its limit is \(e\), Euler's number, approximately 2.71828.

Step 1: Calculate the First Six Terms

Substitute values of \(n\) from 1 to 6 into the sequence \(\left\{a_{n}\right\}=\left\{\left(1+\frac{1}{n}\right)^{n}\right\}\). \nThe terms will be: \n1. For \(n=1\), \(a_1 = \left(1+\frac{1}{1}\right)^1 = 2\) \n2. For \(n=2\), \(a_2 = \left(1+\frac{1}{2}\right)^2 = 2.25\) \n3. For \(n=3\), \(a_3 = \left(1+\frac{1}{3}\right)^3 \approx 2.37\) \n4. For \(n=4\), \(a_4 = \left(1+\frac{1}{4}\right)^4 \approx 2.44\) \n5. For \(n=5\), \(a_5 = \left(1+\frac{1}{5}\right)^5 \approx 2.49\) \n6. For \(n=6\), \(a_6 = \left(1+\frac{1}{6}\right)^6 \approx 2.52\) Note that the final three results have been rounded off to two decimal places.

Step 2: Analyzing Convergence

Observe the terms of the sequence. The numbers appear to be approaching a certain value as n increases. This suggests that the sequence might be convergent.

Step 3: Determine the Limit

It's known that \(\lim_{n\to \infty} \left(1+\frac{1}{n}\right)^{n} = e \approx 2.71828\) (Euler's number). However, the accuracy depends on the limit of \(n\), for example when \(n->\infty\). As \(n\) increases, the terms of the sequence will approach \(e\), which indicates that the sequence is convergent.