Consider the infinite sequence $ \ \ 1,2,4,8,16,...............\ $
Consider the infinite sequence $ \ \ 1,2,4,8,16,............... $
Give the generating function in closed form (i.e., not as an infinite sum and use the most general choice of form for general term of each sequence).
Answer:
I am little confused which one is to be written among the two answer :
(i) generating function $=\sum_{n=0}^{\infty} 2^n x^{n} \ $
(ii) $ a_n=2^n , \ \ n=0,1,2,3,....... $
Someone please respond me about the correct one of the above two options.
$\endgroup$2 Answers
$\begingroup$We have the generating function $$\sum_{n=0}^{\infty} 2^nx^n$$ and are supposed to write it in a closed form. Assuming that $x$ is properly chosen (or not caring about that at all if you are working with formal power series), we can rewrite this as $$\sum_{n=0}^{\infty} 2^nx^n = \sum_{n=0}^{\infty} (2x)^n = \frac{1}{1-2x}.$$
The last one is a closed form and most likely what you are supposed to find.
$\endgroup$ 2 $\begingroup$The generating function is the sum in your choice (1), but they said "in closed form", so you need to use your knowledge of series to rewrite that function of $x$ without an infinite sum.
$\endgroup$ 2
