1. ## sum of series

i hope this is the right place:

Find the sum of:
$\sum_{n=1} ^{10} {3*(\frac{1}{4})^n}$

$=1-(\frac{1}{4})^{10}) = 1$

but there is an additional problem:

Find the error between $S_{10}$ and $s_\infty$. Use the formula for $S_n$ and $s_\infty$ of convergement geometric series , find a general formula that gives the error between $S_n$ and $s_\infty$. Use the new formula to explain why, for the series in #7, the error between $S_n$ and $s_\infty$ is always equal to $r^n$.

2. Originally Posted by Anemori
i hope this is the right place:

Find the sum of:
$\sum_{n=1} ^{10} {3*(\frac{1}{4})^n}$

$=1-(\frac{1}{4})^{10}) = 1$

but there is an additional problem:

Find the error between $S_{10}$ and $s_\infty$. Use the formula for $S_n$ and $s_\infty$ of convergement geometric series , find a general formula that gives the error between $S_n$ and $s_\infty$. Use the new formula to explain why, for the series in #7, the error between $S_n$ and $s_\infty$ is always equal to $r^n$.

$\sum_{k=0}^nar^k=a\frac{1-r^{n+1}}{1-r}$
$\sum_{k=0}^\infty ar^k=\frac{a}{1-r}$
Since we are dealing with sums starting with $k=1$, we'll need to subtract that out.
$S_n=\sum_{k=1}^nar^k=a\frac{1-r^{n+1}}{1-r}-a$
$s_\infty = \sum_{k=1}^\infty ar^k=\frac{a}{1-r}-a$
The term error is used because the sequence $S_1, S_2, S_3, \cdots$ can be seen as approximations to $s_\infty$ that get closer and closer as $n$ increases. The error is the true value minus the value of the approximation, or $s_\infty-S_n$.