Geometric Sequence Sum with Non-Traditional First Term?

[kylera]

In words, the sum of a geometric sequence can be written out to say "the first term divided by (1 minus the common ratio)". Does the first term also apply when the series starts with some other number n other than 1 (like 2 or 3, etc)? In other words, the first term is when n = some other number instead of 1.

 

[HallsofIvy]

Why in the world would you even ask?

$$\sum_{n=0}^\infty 1/2^n= 1+ 1/2 + 1/4+ \cdot\cdot\cdot$$
is a geometric series that sums to
[tex]\frac{1}{1- 1/2}= 2[/itex]
Why would you think that
$$\sum_{n=1}^\infty 1/2^n= 1/2 + 1/4+ \cdot\cdot\cdot$$
sums to the same thing? It is missing the initial 1 so it sums to 2-1= 1.
Similarly
$$\sum_{n=2}^\infty 1/2^n= 2- 1- 1/2= 1/2$$
and
$$\sum_{n= 3}^\infty 1/2^n= 2- 1- 1/2- 1/4= 1/4$$

 

[kylera]

Well, I'm sorry if the question sounded silly and amateurish, but the book I'm using didn't emphasize that aspect.