Final answer:
To determine how many terms of a convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude, we can use the remainder formula for geometric series.
Step-by-step explanation:
To determine how many terms of a convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude, we can use the concept of the remainder formula for geometric series.
The remainder formula for a geometric series is given by:
R = a * (1 - r^n) / (1 - r)
In this formula, a is the first term, r is the common ratio, n is the number of terms summed, and R is the remainder.
To find the number of terms needed, we can rearrange the formula:
n = log((R * (1 - r)) / a) / log(r)
By substituting the known values into the formula and solving for n, we can find the number of terms needed.