数列极限入门习题

数列极限入门习题

1. $\underset{n\to \infty }{\lim }(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}{)}^{\frac{1}{n}}$
2. $\underset{n\to \infty }{\lim }(\frac{1}{n+\sqrt{1}}+\frac{1}{n+\sqrt{2}}+\cdots +\frac{1}{n+\sqrt{n}})$
3. $\underset{n\to \infty }{\lim }\sum _{k=n^2}^{(n+1{)}^2}\frac{1}{\sqrt{k}}$
4. $\underset{n\to \infty }{\lim }\frac{1\cdot 3\cdot 5\cdot \cdots \cdot (2n-1)}{2\cdot 4\cdot 6\cdot \cdots \cdot (2n)}$
5. $\underset{n\to \infty }{\lim }\frac{3n^2+4n-1}{n^2+1}$
6. $\underset{n\to \infty }{\lim }\frac{n^3+2n^2-3n+1}{2n^3-n+3}$
7. $\underset{n\to \infty }{\lim }\frac{3^n+n^3}{3^{n+1}+(n+1{)}^3}$
8. $\underset{n\to \infty }{\lim }(\sqrt[n]{n^2+1}-1)\sin \frac{n\pi }{2}$
9. $\underset{n\to \infty }{\lim }\sqrt{n}(\sqrt{n+1}-\sqrt{n})$
10. $\underset{n\to \infty }{\lim }\sqrt{n}(\sqrt[4]{n^2+1}-\sqrt{n+1})$
11. $\underset{n\to \infty }{\lim }\sqrt[n]{\frac{1}{n!}}$
12. $\underset{n\to \infty }{\lim }(1-\frac{1}{2^2})(1-\frac{1}{3^2})\cdots (1-\frac{1}{n^2})$
13. $\underset{n\to \infty }{\lim }\sqrt[n]{n\lg n}$
14. $\underset{n\to \infty }{\lim }(\frac{1}{2}+\frac{3}{2^2}+\cdots +\frac{2n-1}{2^n})$
15. 已知$\underset{n\to \infty }{\lim }{a}_n=a$，$\underset{n\to \infty }{\lim }{b}_n=b$，证明：
    $$\underset{n\to \infty }{\lim }\frac{a_1{b}_n+a_2{b}_{n-1}+\cdots +a_n{b}_1}{n}=ab$$