Question

1. What does it mean for a sequence {a} to converge to a € R? State the definition. (-1)n+1 2. Prove that lim = 0 n 2n 3. Prove that lim +0n + 1 = 2 80 4. Prove that lim +-+V5n 9 - 7 5. Prove that lim 108 + 137 13

Answer 1

1) The sequence is said to be convergent to $a\in R$ if given $\epsilon >0$ we can find a M such that

$|a_n-a|<\epsilon ~~~~~~~~~~\text{ for all }n>M$

2) Since

$\left |{ (-1)^{n+1} \over n} -0\right |= \left |{ (-1)^{n+1} \over n} \right |={1\over n}< \epsilon ~~~~~\text{ for all } n>M$

Say if we take $\epsilon =0.001$ then M=1001

Hence $\lim_{n \rightarrow \infty} {(-1)^{n+1}\over n}=0$

3) Since

$\left |{ 2n \over n+1} -2\right |= \left |{ -2 \over n+1} \right |={2\over n+1}< \epsilon ~~~~~\text{ for all } n>M$

Say if we take $\epsilon =0.001$ then M=1000

Hence, $\lim_{n\rightarrow \infty}{ 2n \over n+1} =2$

4) Since

$\left |{ 80 \over \sqrt{5n}} -0\right |= \left |{ 80 \over \sqrt{n}} \right |={ 80 \over \sqrt{n}} < \epsilon ~~~~~\text{ for all } n>M={6400\over \epsilon^2}$

Hence, $\lim_{n\rightarrow \infty}{ 80 \over \sqrt{n}} =0$