I have a couple questions about the sums of geometric series.
One. So the formula for the sum is
t(n)=t(1)[(r^n)-1]
But if my series starts at t(0), can I change the formula to
t(n)=t(0)[(r^n)-1]
?
Two. If, in the series, there is a different pattern in the numerator than in the denominator, how can I solve it?
Such as: [x/2]+[(x^2)/4]+[(x^3)/8]...
Thank you.
One. The formula you mentioned, t(n) = t(1)[(r^n)-1], is the formula for the sum of a geometric series that starts at t(1). If your series starts at t(0), you will need to modify the formula to take that into account.
To derive the formula for a geometric series starting at t(0), you can think of it as a translation of the series that starts at t(1). In other words, the term t(0) is equivalent to t(1) in the original series, t(1) is equivalent to t(2), and so on.
To get the formula for a series starting at t(0), you can shift the formula by one term backwards. You can do this by using the following formula:
t(n) = t(1)[(r^n)-1] / (r - 1)
This modified formula takes into account the fact that t(0) is equivalent to t(1) in the original formula.
Two. In your example, you have a series with a different pattern in the numerator and denominator. To solve it, you can rewrite each term of the series in a way that the pattern becomes clear.
For [x/2], the pattern in the numerator is x raised to the power of 1, and in the denominator, it is 2 raised to the power of 1. Similarly, for [(x^2)/4], the numerator has x raised to the power of 2, and the denominator has 2 raised to the power of 2.
To find the sum of this series, you can rewrite each term in a common form by choosing a common base for both the numerator and denominator. In this case, you can choose 2 as the common base.
Rewriting each term:
[x/2] = (x/2^1)
[(x^2)/4] = (x^2/2^2)
[(x^3)/8] = (x^3/2^3)
Now that each term is in a familiar form, you can see that the pattern in the numerators is x raised to the power of 1, 2, 3, and so on, and the pattern in the denominators is 2 raised to the power of 1, 2, 3, and so on.
So, the series can be rewritten as:
(x/2^1) + (x^2/2^2) + (x^3/2^3) + ...
This new form allows you to see that the pattern is now a geometric series with the first term being (x/2), the common ratio being (1/2), and the exponent of the common ratio increasing by 1 in each term.
With this new form, you can now use the formula for the sum of a geometric series to find the sum of this particular series.