Big theta for a S(n)

Question

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.

score 0 · Answer 1 · answered Oct 02 '13 at 05:08

0

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 $S(n) < \dfrac1{1-c}$.

answered Oct 02 '13 at 05:08

marty cohen

107,799

ok but how did you know to compare S(n) to those two constants (how did you know what constants to use)? – notamathwiz Oct 02 '13 at 05:20

Big theta for a S(n)

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