Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.R.1c

Chapter 10, Problem 10.R.1c

Determine whether the following statements are true and give an explanation or counterexample.
c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.

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Recall that the sequence of partial sums \( S_n = \sum_{k=1}^n a_k \) represents the sum of the first \( n \) terms of the series \( \sum a_k \).

If the terms of the sequence of partial sums \( S_n \) approach a limit \( L \) as \( n \to \infty \), then the infinite series \( \sum a_k \) converges to \( L \).

In this problem, it is given that the terms of the sequence of partial sums approach \( \frac{5}{2} \). This means \( \lim_{n \to \infty} S_n = \frac{5}{2} \).

Since the partial sums approach \( \frac{5}{2} \), by definition, the infinite series \( \sum a_k \) converges and its sum is \( \frac{5}{2} \).

Therefore, the statement is true because the convergence of the partial sums to \( \frac{5}{2} \) directly implies the series converges to \( \frac{5}{2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sequence of Partial Sums

The sequence of partial sums is formed by adding the first n terms of a series. It helps analyze the behavior of the series by examining the limit of these sums as n approaches infinity.

Convergence of an Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. This limit is the sum of the series, meaning the series adds up to a specific value.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence get arbitrarily close to as the index grows large. If the partial sums approach 5/2, the series converges to 5/2.

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