The series can be written
which is clearly a geometric series with first term 2 and a common ratio of 2x.
Therefore, the sum is
.
I am having a bit of difficulty in determining the sum of an infinite geometric series:
Find the sum (assume ):
I was shown that the sum of this series is equal to the following when :
Where 2 is the first value. But I thought that since this series was increasing by a common ratio of 2x that it really depends upon how small or large x really is. This is where I am stuck. I am just not sure how to go around the lack of a definitive x value. I appreciate the help.
Ah, woops. Thank you, I just didn't realize that you could leave this in terms of X. I appreciate that, but I also have one more question, sort of revolving around the same concept, so I will post it below.
Find the sum of:
Would the sum of this simply be:
I just wanted to say thanks to everyone that has helped me out. I had missed class yesterday and this has helped make it very clear. Thanks again.
EDIT:
Sorry for asking so many questions about this, I just have two more questions regarding closed form formulas for geometric series and summing infinite series.
For the closed form formula, I have an expression for the term:
But I am not sure how to convert this to closed form since I am not really sure what closed form looks like. Is it just to convert it into a finite sum equation like this:
For the infinite series, I have the formula:
Where
So the very first term would be:
And this would go in the numerator, but I am not sure what would go in the denominator. Again, I really appreciate the help thus far.