what is an upper bound on the absolute value of the error?
the abs value of (the sum from n=1 to infinity of ((-1)^(n+1))/(n*5^n) where it is -.1826666....)
Math Homework Help: $1/Question
Search for us on google
To find an upper bound on the absolute value of the error, we need to calculate the absolute difference between the given series and its closest convergent value.
The given series is:
S = (-1)^(n+1) / (n * 5^n)
Let's calculate the sum of the series to get an approximate value.
By evaluating the series for a certain number of terms, we can get a partial sum. The more terms we consider, the more accurate the approximation becomes. Let's calculate the partial sums for different numbers of terms:
For n = 1 to 10: S10 = (-1/5) + (1/50) + (-1/500) + (1/5000) + ...
For n = 1 to 100: S100 = (-1/5) + (1/50) + (-1/500) + (1/5000) + ...
And so on...
By adding more terms, we can see that the partial sums (Sn) approach a certain value. In this case, that value is approximately -0.1826666.
Now, to find an upper bound on the absolute value of the error, we can compare the exact value (S) with the closest convergent value (Sn). In this case, we can use S10 as an approximation.
The absolute value of the error is given by |S - Sn|.
Let's calculate it:
|S - S10| = |-0.1826666 - S10|
By calculating this difference, we can get an upper bound on the absolute value of the error.