Analysis & Series Homework #1 & #2

[erikalculator]

Homework Statement

Number 1.

a_n=(1+1/n)ⁿ for all n
b_n=(1-1/n)^-n n=2,3,4,...

i) Prove that 1<b_n/a_n<=n/(n-1) n=2,3,4...
ii)Show that <b_n> also converges to e'. "e" being the exponential.

Number 2.

i) Suppose that 1<= r <= n for all r and all n. Prove 1 + series from m=1 to r [(1/m!)*(1-1/n)*(1-2/n)...(1-(m-1)/n)] <= a_n <= e.

ii) Prove that 1 + series from m=1 to r [1/m!] <= e' <=e.

iii) prove that e'=e.number 2 I understand conceptually it makes logical sense but I am not sure how to prove it.

 

[silvermane]

Homework Statement

Number 1.

a_n=(1+1/n)ⁿ for all n
b_n=(1-1/n)^-n n=2,3,4,...

i) Prove that 1<b_n/a_n<=n/(n-1) n=2,3,4...
ii)Show that <b_n> also converges to e'. "e" being the exponential.

Number 2.

i) Suppose that 1<= r <= n for all r and all n. Prove 1 + series from m=1 to r [(1/m!)*(1-1/n)*(1-2/n)...(1-(m-1)/n)] <= a_n <= e.

ii) Prove that 1 + series from m=1 to r [1/m!] <= e' <=e.

iii) prove that e'=e.


number 2 I understand conceptually it makes logical sense but I am not sure how to prove it.

Maybe try using the conjugate of the denominator and multiplying that by the top and bottom of your fraction. I think that could help, but I also think that there may be more information needed to solve this. :(