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Calculus IIChapter 4: Series and SequencesSection 4-1

Sequences — Question 4

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Question 4

Let an=(1+2n)na_n=\left(1+\dfrac2n\right)^n.

  1. Interpret ana_n as the balance produced by compounding a principal of 11 at a total rate of 22 over one time unit, using nn periods.

  2. Evaluate lim⁡n→∞an\lim_{n\to\infty}a_n using the standard exponential limit.

  3. Give a decimal approximation of the limiting balance.

Original worksheet page 1: question and worked solution for 4-1-004
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Question 4 – Solution

Step 1: Interpret the compounding model.

The principal is P=1P=1. Dividing the total rate 22 into nn equal periods gives rate 2/n2/n per period; applying that growth factor nn times produces 1(1+2/n)n1(1+2/n)^n.

Step 2: Rewrite in the standard exponential form.

Put m=n/2m=n/2. Then m→∞m\to\infty and an=[(1+1m)m]2.a_n=\left[\left(1+\frac1m\right)^m\right]^2.

Step 3: Apply the standard limit.

Using lim⁡m→∞(1+1/m)m=e\lim_{m\to\infty}(1+1/m)^m=e and continuity of the squaring function gives limn→∞an=e2≈7.38906.\lim_{n\to\infty}a_n=e^2\approx7.38906.

Step 4: Interpret and verify.

This agrees with the continuous-compounding formula Pert=e2Pe^{rt}=e^2 for P=1P=1, r=2r=2, and t=1t=1. The standard exponential limit supplies the proof.

Original worksheet page 2: question and worked solution for 4-1-004

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