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

Name: Problem 96, Chapter 11: Calculus: Early Transcendentals
Uploaded: 2021-12-23T07:26:35-08:00
Duration: 1 min 13 s
Channel: Manik Pulyani

step 1 Now for this question we have clearly we have to show that 0 is less than A and A is greater tha

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

Let $a_n=(1+1 / n)^n$. (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 $e$. See Equation 3.6.6.)

Let $a_n=(1+1 / n)^n$.
(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 $e$. See Equation 3.6.6.)