41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

Verified step by step guidance

1. Identify the series given: it is a geometric series with the general term \( a_k = \frac{1}{3^k} \) for \( k = 1, 2, 3, \ldots \).
2. Recall the formula for the sum of an infinite geometric series starting at \( k=1 \): \[ S = \frac{a_1}{1 - r} \] where \( a_1 \) is the first term and \( r \) is the common ratio.
3. Calculate the first term \( a_1 = \frac{1}{3^1} = \frac{1}{3} \) and the common ratio \( r = \frac{1}{3} \).
4. Find the partial sum of the first 10 terms using the formula for the sum of the first \( n \) terms of a geometric series: \[ S_{10} = a_1 \frac{1 - r^{10}}{1 - r} \]. This gives the approximation of the series using 10 terms.
5. Use the remainder estimate for a geometric series to find the interval where the true sum lies: the remainder \( R_{10} = S - S_{10} = \frac{a_1 r^{10}}{1 - r} \). Since the series is positive and decreasing, the true sum lies between \( S_{10} \) and \( S_{10} + R_{10} \).

Key Concepts

Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. For |r| < 1, the infinite series converges to a finite sum S = a / (1 - r), where a is the first term. Understanding this helps identify the sum of the given series ∑ 1/3^k.

Partial Sums and Approximation

A partial sum is the sum of the first n terms of a series and serves as an approximation to the infinite sum. Using partial sums allows us to estimate the series value by truncating after a finite number of terms, which is essential when calculating the sum up to ten terms.

Remainder (Error) Estimation in Geometric Series

The remainder after n terms in a geometric series is the difference between the infinite sum and the partial sum. It can be calculated as R_n = S - S_n = a * r^n / (1 - r), providing an interval where the true sum lies. This concept is crucial for bounding the series value after ten terms.