Determine if the limits exist
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.8 Solution: Part aWe have $\lim n^3=\infty$. Part bWe have $\lim(-n^3)=-\infty$. Part cWe have $\lim(-n)^n$ does not…
Show the limit is zero
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.7 Solution: We have seen that $0\le s_n<\sqrt{\dfrac{2}{n-1}}$. By Sequeeze-Theorem/Exercise 8.5, it suffices to show that $\lim…
Finding the limit by solving equation needs convergence
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.6 Solution: Part a This part is the same as that of Exercise 9.4 and Exercise 9.5.…
Find the limit by solving equation II
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.5 The idea is pretty much the same as Exercise 9.4. Solution: Since $(t_n)$ converges. Set $\lim…
Find the limit by solving equations
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.4 Solution: Part a $s_1=1$, $s_2=\sqrt{2}$, $s_3=\sqrt{\sqrt 2+1}$, $s_4=\sqrt{\sqrt{\sqrt 2+1}+1}$. Part b Since $(s_n)$ converges. Set $\lim…
Show limit using limit theorems carefully
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.3 Solution: By Theorem 9.4, we have$$\lim a_n^3=(\lim a_n)^3=a^3,\quad \lim b_n^2=(\lim b_n)^2=b^2$$ By Theorem 9.2, we have$$\lim…
Limit of sum is sum of limits
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.2 Solution: Part aBy Theorem 9.3, we have$$\lim(x_n+y_n)=\lim x_n+\lim y_n=3+7=10.$$ Part bBy Theorems 9.2 and 9.3 we…
Justify the following limits
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 9 Exercise 9.1 Solution: Part aLet $s_n=\dfrac{n+1}{n}$. We have$$s_n=1+\frac{1}{n}.$$ By Theorem 9.7(a) we have $\lim\dfrac{1}{n}=0$. Thus by Theorem 9.3,…
If the limit is greater than a number, so are all but finitely many of numbers in this sequence
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 8 Exercise 8.10 This part actually has been shown in Exercise 8.9. Solution: Suppose $\lim s_n=s$. Then we have…
If the sequence is greater than a number, so is the limit
Solution to Elementary Analysis: The Theory of Calculus Second Edition Section 8 Exercise 8.9 Solution: Part aWe argue it by contradiction. Suppose $\lim s_n=s < a$. Then $a-s >0$. Since…