Let an=(1+(1)/(n))^n (a) Show that if 0 ⩽a<b, then (b^n+1-a^n+1)/(b-a)<(n+1) b^n (b) Deduce that

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Let an=(1+(1)/(n))^n (a) Show that if 0 ⩽a<b, then...

Let $a_{n}=\left(1+\frac{1}{n}\right)^{n}$ (a) Show that if $0 \leqslant a<b,$ then $\frac{b^{n+1}-a^{n+1}}{b-a}<(n+1) b^{n}$ (b) Deduce that $b^{n}[(n+1) a-n b]<a^{n+1}$ (c) Use $a=1+1 /(n+1)$ and $b=1+1 / n$ in part (b) to show that $\left\{a_{n}\right\}$ is increasing. (d) Use $a=1$ and $b=1+1 /(2 n)$ in part (b) to show that $a_{2 n}<4$. (e) Use parts (c) and (d) to show that $a_{n}<4$ for all $n .$ (f) Use Theorem 12 to show that $\lim _{n \rightarrow \infty}(1+1 / n)^{n}$ exists. (The limit is $e . \text { See Equation } 3.6 .6 .)$

Let $a_{n}=\left(1+\frac{1}{n}\right)^{n}$ (a) Show that if $0 \leqslant a<b,$ then $\frac{b^{n+1}-a^{n+1}}{b-a}<(n+1) b^{n}$
(b) Deduce that $b^{n}[(n+1) a-n b]<a^{n+1}$
(c) Use $a=1+1 /(n+1)$ and $b=1+1 / n$ in part (b) to show that $\left\{a_{n}\right\}$ is increasing.
(d) Use $a=1$ and $b=1+1 /(2 n)$ in part (b) to show
that $a_{2 n}<4$.
(e) Use parts (c) and (d) to show that $a_{n}<4$ for all $n .$
(f) Use Theorem 12 to show that $\lim _{n \rightarrow \infty}(1+1 / n)^{n}$ exists. (The limit is $e . \text { See Equation } 3.6 .6 .)$

Key Concepts

Exponential Sequence Approximating e

This concept involves the sequence defined by a? = (1 + 1/n)? whose limit as n approaches infinity is the constant e. Understanding this sequence is central to many areas of analysis, particularly in defining the exponential function and continuous growth processes.

Monotonic Sequences

A sequence is said to be monotonic if it is either entirely nonincreasing or nondecreasing. In this context, showing that the sequence is increasing (monotonic) is crucial since monotonic sequences have well-behaved limits, which aids in proving convergence.

Bounded Sequences

A bounded sequence is one that stays within fixed upper and lower limits. The concept is important here because when a sequence is both bounded and monotonic, the Monotone Convergence Theorem guarantees that the sequence converges.

Mean Value Theorem and Inequalities

The Mean Value Theorem is a fundamental tool in calculus that can be used to derive inequalities relating to functions. In the problem, it is implicitly used in establishing bounds for expressions involving powers, which aids in comparing different terms of the sequence.

Algebraic Manipulation of Exponential Inequalities

Manipulating expressions that involve exponents and inequalities is key to many analysis proofs, including those that establish monotonicity and bounds. This concept includes techniques for handling expressions like (1 + 1/n)? and transforming inequalities to deduce limits and other properties.

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