极限和证明数列收敛的示例

Finding Limits of Sequences Algebraically

In many cases, such as $\lim_{n\to\infty} 2^{1/n}$, we can interpret a sequence limit as a function limit towards infinity. Sometimes we just need to reason directly, like for $\lim_{n\to\infty} (1+x+x^2+\dots+x^n) = 1+x+x^2+\dots$ (infinite geometric series as a limit), using summation or other algebraic manipulation first.