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


c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

Verified step by step guidance

1. Recognize that the series given is a geometric series of the form \(\sum_{k=1}^{\infty} \frac{1}{3^k}\), where the first term \(a = \frac{1}{3}\) and the common ratio \(r = \frac{1}{3}\).
2. Recall the formula for the sum of an infinite geometric series when \(|r| < 1\): \(S = \frac{a}{1 - r}\). This gives the exact sum of the series.
3. To find the lower and upper bounds \(L_n\) and \(U_n\) for the partial sum \(S_n = \sum_{k=1}^n \frac{1}{3^k}\), calculate the partial sum using the finite geometric series formula: \(S_n = a \frac{1 - r^n}{1 - r}\).
4. Understand that the remainder (or error) after \(n\) terms, \(R_n = S - S_n\), can be expressed as \(R_n = \frac{a r^n}{1 - r}\). This remainder helps in estimating how close \(S_n\) is to the exact sum \(S\).
5. Use the partial sum \(S_n\) and the remainder \(R_n\) to establish the bounds: the lower bound \(L_n = S_n\) and the upper bound \(U_n = S_n + R_n\). This means the exact sum \(S\) lies between \(L_n\) and \(U_n\).

Key Concepts

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. For |r| < 1, the infinite geometric series converges to a finite sum S = a / (1 - r), where a is the first term. Understanding this helps evaluate the exact sum of series like ∑ 1/3^k.

Partial Sums and Remainders

Partial sums are the sums of the first n terms of a series, providing an approximation to the infinite sum. The remainder (or tail) is the difference between the infinite sum and the partial sum. Estimating remainders helps find bounds on the exact value of the series.

Bounds on Series Sums

For convergent series, the exact sum lies between the partial sum and the partial sum plus the remainder estimate. Using remainder estimates, one can find lower and upper bounds (Lₙ and Uₙ) to bracket the true sum, ensuring an accurate approximation within a known error margin.