Question

Note the following sequence:

The question is, for which complex z values does it converge for n->infinity, giving limits, and for which complex z values does it diverge to infinity in C?

For which complex values of z (if any) do the following sequences converge as n → ∞ (give the limits when they do) and for which complex values of z (if any) do they diverge to ∞ in C?

Heres the sequence

an(z)=(1+(z/n))^n

Answer 1

Given Sequence :

2n

So $\lim_{z\rightarrow\infty }z_n=\left ( 1+\frac{z}{n} \right )^{n}=e^{z}$ (its formula)

  $=e^{a+ib}=e^{a}e^{ib}$ (z=a+ib, a, b are real)

so for unique value for b=0, i. e. imaginary part must be zero.

So,   i. e. limit exist.

linn:n = (1 + n). e.

Hence, for limit to be exist z must be real i. e. z belongs to set of real numbers. And it diverges or diverges to infinity when z is not real.