- #1

MathematicalPhysicist

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## Main Question or Discussion Point

i need to prove that the sequence is increasing, i.e, x_n+1>=x_n

which translates into:

[(n+2)/(n+1)]^(n+1)>=[(n+1)/n]^n

i tried through induction but got stumbled:

n=k

[(k+2)/(k+1)]^(k+1)>= [(k+1)/k]^k

n=k+1

[(k+3)/(k+2)]^(k+2)>=[(k+2)/(k+1)]^(k+1)

now i need to prove the last inequality, and got baffled.

any help is appreciated, perhaps induction isn't the right way?

p.s

i know this sequence converges to e, so spare me the trivial details about this sequence.

thanks in advance.

which translates into:

[(n+2)/(n+1)]^(n+1)>=[(n+1)/n]^n

i tried through induction but got stumbled:

n=k

[(k+2)/(k+1)]^(k+1)>= [(k+1)/k]^k

n=k+1

[(k+3)/(k+2)]^(k+2)>=[(k+2)/(k+1)]^(k+1)

now i need to prove the last inequality, and got baffled.

any help is appreciated, perhaps induction isn't the right way?

p.s

i know this sequence converges to e, so spare me the trivial details about this sequence.

thanks in advance.