What is an upper bound on the absolute value of the error:
|sum from n=1 to infinity of ((-1)^(n+1))/(n*5^n)) -.1826666...|
?????
To find an upper bound on the absolute value of the error, we need to determine how close the partial sums of the given series get to the actual value of the series.
The given series is an alternating series, (-1)^(n+1)/(n*5^n), with alternating signs and decreasing absolute values.
To find an upper bound on the error, we can use the Alternating Series Estimation Theorem, which states that the error of an alternating series is less than or equal to the absolute value of the first omitted term.
In this case, the first omitted term is the term with n = 1. Plugging in n = 1 into the series expression, we get:
|(-1)^(1+1)/(1*5^1)| = |1/5|
So, an upper bound on the absolute value of the error is 1/5, or 0.2.
Therefore, |sum from n=1 to infinity of ((-1)^(n+1))/(n*5^n)) - 0.1826666...| ≤ 0.2.