Calculus II

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You need to consider the following sequences, such that:

`(a_n) = (1+7)*(1 + 7/2)^2*(1+7/3)^3*(1+7/4)^4*...*(1+7/n)^n`

`(b_n) = (1+7/n)^n*(1 + 7/n)^n*(1+7/n)^n*(1+7/n)^n*...*(1+7/n)^n = n*(1+7/n)^n`

You need to notice that `(a_n) < (b_n)` , hence `lim_(n->oo) (a_n) < lim_(n->oo) (b_n)`

You need to evaluate ` lim_(n->oo) (b_n) ` such that:

`lim_(n->oo) (b_n) = lim_(n->oo)...

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You need to consider the following sequences, such that:

`(a_n) = (1+7)*(1 + 7/2)^2*(1+7/3)^3*(1+7/4)^4*...*(1+7/n)^n`

`(b_n) = (1+7/n)^n*(1 + 7/n)^n*(1+7/n)^n*(1+7/n)^n*...*(1+7/n)^n = n*(1+7/n)^n`

You need to notice that `(a_n) < (b_n)` , hence `lim_(n->oo) (a_n) < lim_(n->oo) (b_n)`

You need to evaluate ` lim_(n->oo) (b_n) ` such that:

`lim_(n->oo) (b_n) = lim_(n->oo) (n*(1+7/n)^n)`

`lim_(n->oo) (n*(1+7/n)^n) = lim_(n->oo) (n)* lim_(n->oo)((1+7/n)^n)`

`lim_(n->oo) (n*(1+7/n)^n) = lim_(n->oo) (n)*lim_(n->oo)(((1+7/n)^1/(7/n)))^((7/n)*n)`

`lim_(n->oo) (n*(1+7/n)^n) = lim_(n->oo) (n)*e^ lim_(n->oo)((7/n)*n)`

`lim_(n->oo) (n*(1+7/n)^n) = lim_(n->oo) (n)*(e^7)`

`lim_(n->oo) (n*(1+7/n)^n) = e^7*lim_(n->oo) (n)`

`lim_(n->oo) (n*(1+7/n)^n) = oo`

Since `lim_(n->oo) (b_n) = oo` and `lim_(n->oo) (b_n) > lim_(n->oo) (a_n)` , hence `lim_(n->oo) (a_n) = lim_(n->oo)((1+7)*(1 + 7/2)^2*(1+7/3)^3*(1+7/4)^4*...*(1+7/n)^n) = oo.`

Hence, evaluating if the sequence converges or diverges yields that the sequence `a_n = ((1+7)*(1 + 7/2)^2*(1+7/3)^3*(1+7/4)^4*...*(1+7/n)^n` diverges.

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