Assuming that the series is geometric, find the sum (correct up to 8 decimal places) of the first 25 terms of the following series:
$\displaystyle 1-\frac{1}{2}+\frac{1}{4}-...$
Thanks
jpatrie
The sum of a geometric series through the first n terms, if the first term is one, is $\displaystyle \frac{1 - r^n}{1 - r}$, where r is the common ratio. In your example, r = -1/2. So the sum is $\displaystyle \frac{1 - (-\frac{1}{2})^{25}}{1 + \frac{1}{2}}$.
It is geometric. The common ratio is $\displaystyle -\frac{1}{2}$. Since $\displaystyle |r|<1$, it is also an infinite geometric series... but that's another story. $\displaystyle a=1$.
The formula for sum of a geometric sequence is
$\displaystyle S_n = \frac{a(1-r^n)}{1-r}$.
We want the sum of the first 25 terms... so we have
$\displaystyle S_{25} = \frac{1\left(1 - \left(-\frac{1}{2})^{25}\right)\right)}{1-\left(-\frac{1}{2}\right)} = \frac{1 + \frac{1}{2^{25}}}{\frac{3}{2}}$.