I'm trying to calculate $e$ to a certain number of digits. The Maclaurin Series expansion of $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. When $x = 1$ we can approximate the value of $e$ by evaluating $\displaystyle\sum_{n=0}^m\frac1{n!}$. How can I find the error in the approximation when I compute the sum to $m$? If I take the output from the series, how many digits (truncated) of my approximation to $e$ are correct if I evaluate it to $m$ terms?
In general, how can we find the error when evaluating such a series?
Another good series, Brother's Formula (converges to $e$ faster than the one noted earlier):
$$\sum_{n=0}^\infty \frac{2n+2}{(2n+1)!}$$
Edit: To find the digits that are correct, add the upper bound for the error to the estimate and the digits before the first digit that changed are correct. For example, let's say that we have found $3.1234122$ as an estimate to some series. Let's say that we compute the error to be less than $0.0001879$. To find the digits that are correct in the estimate:
$$3.1234122 + 0.0001879 = 3.1236001$$
Thus we can see that the digits that we can be sure to be correct are $3.123$ so the first 4 digits of the number are $3.123$.
Best Answer
There is a simple way to bound the error. Suppose you did your aproximation until some $m$, I mean, you computed $\displaystyle\sum_{n=0}^m\frac{1}{n!}$. Your exact error is $\displaystyle\sum_{n=0}^\infty\frac{1}{n!}-\sum_{n=0}^m\frac{1}{n!}=\sum_{n=m+1}^\infty\frac{1}{n!}$.
But note that $$\sum_{n=m+1}^\infty\frac{1}{n!}<\sum_{n=1}^\infty\frac{1}{(m+1)^n}=\frac{1}{m}.$$ What is an acceptable(if the idea is just to get some bound, not necessarily a good bound) error bound.