can anyone help me please,thanks><

[chuang tsai-ling]

1. lim(n→infinity) 　nsin(1/2n)

2.Determine whether the given sequences｛a_n｝ converge,diverge to infinity,or oscillate.If the limit exists,prove it.
a_n=(n^n)/n!

3.Suppose that the sequences {a_{n} converges to A. Define the sequences {bn} by bn = (}a_n+a_{n+1)/2.
Does the sequences {bn} coverges?If so,specify the limit and prove ur conclusion. Otherwise ,give an example when this is not true.}



These three questions is my homework,but I don't have any idea how to solve,thanks to help me.

 

[HallsofIvy]

1. lim(n→infinity) 　nsin(1/2n)

Let m= 1/2n. As n goes to infinity, m goes to 0.

2.Determine whether the given sequences｛a_n｝ converge,diverge to infinity,or oscillate.If the limit exists,prove it.
a_n=(n^n)/n!

n factorial is a product of n terms, from 1 to n. n^n is a product of n terms all equal to n. Which is larger?

3.Suppose that the sequences {a_n} converges to A. Define the sequences {bn} by bn = (a_n+a_n+1)/2.
Does the sequences {bn} coverges?If so,specify the limit and prove ur conclusion. Otherwise ,give an example when this is not true.

If sequence \(\displaystyle \{a_n\}\) converges to A then \(\displaystyle \{a_{n+1}\}\) also converges to A.

[/SUB][/I]These three questions is my homework,but I don't have any idea how to solve,thanks to help me.

 

[lookagain]

1. lim(n→infinity) 　nsin(1/2n)

chuang tsai-ling, if you meant to type the equivalent of \(\displaystyle nsin\bigg(\dfrac{1}{2n}\bigg), \)

then you must include grouping symbols around the denominator when typing it out horizontally.

Let m= 1/2n. As n goes to infinity, m goes to 0.

HallsofIvy, the same is true for you. What you have is the same as \(\displaystyle \ "Let \ \ m = \frac{1}{2}n."\)