Consider the following function: $$S(n)=1+c+c^2+⋅⋅⋅+c^n,$$ where c is a positive real number.
(A) This function is the sum of a geometric series. Give a precise closed-form formula for S(n), interms of c and n, in the case where c≠1.
which is $S(n)=\frac{c^{n+1}-1}{c-1}$
(B) Show that S(n) is:
θ(1) if c<1
θ(n) if c=1
im having problems solving these two big theta problems can someone give me a nudge in the right direction (i know how big theta is defined), so i assume that i need to find two constants for which the function is bounded, but how do i go about doing this.
First, if $c=1$, $S(n)$ should be pretty obvious.
Second, if $c < 1$, write the formula in the form $S(n) = \dfrac{1-c^{n+1}}{1-c}$ and show that $S(n) > 1$ and $S(n) < \dfrac1{1-c}$.