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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.2.1
Chapter 10, Problem 10.2.1

Finding nth Partial Sums
In Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.
2 + (2/3) + (2/9) + (2/27) + … + (2 / 3ⁿ⁻¹) + …

Verified step by step guidance
1
Identify the type of series given. Notice that the series is 2 + \(\frac{2}{3}\) + \(\frac{2}{9}\) + \(\frac{2}{27}\) + \(\dots\), which is a geometric series where each term is multiplied by \(\frac{1}{3}\) to get the next term.
Write the general term of the series. The first term a is 2, and the common ratio r is \(\frac{1}{3}\). So, the nth term can be expressed as a_n = 2 \(\times\) \(\left\)(\(\frac{1}{3}\)\(\right\))^{n-1}.
Use the formula for the nth partial sum of a geometric series: S_n = a \(\times\) \(\frac{1 - r^n}{1 - r}\). Substitute a = 2 and r = \(\frac{1}{3}\) into this formula to get S_n.
Simplify the expression for S_n by plugging in the values and rewriting the formula clearly: S_n = 2 \(\times\) \(\frac{1 - \left(\frac{1}{3}\)\(\right\))^n}{1 - \(\frac{1}{3}\)}.
Determine if the series converges by examining the limit of S_n as n approaches infinity. Since |r| = \(\frac{1}{3}\) < 1, the series converges, and the sum to infinity is S = \(\frac{a}{1 - r}\).

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Key Concepts

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

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. In this problem, the series has a common ratio of 1/3, making it geometric. Understanding this helps in finding a general formula for the nth partial sum.
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Geometric Series

Nth Partial Sum Formula

The nth partial sum of a geometric series is given by S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. This formula allows calculation of the sum of the first n terms, which is essential for analyzing convergence and total sums.
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Convergence of Infinite Series

An infinite geometric series converges if the absolute value of the common ratio is less than 1. When it converges, the sum approaches S = a / (1 - r) as n approaches infinity. Recognizing convergence is key to determining the series’ total sum.
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Related Practice
Textbook Question

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) sin (1/n)

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Textbook Question

Recursively Defined Sequences

In Exercises 101–108, assume that each sequence converges and find its limit.

a₁ = 5,aₙ₊₁ = √(5aₙ)

Textbook Question

In Exercises 37–42, find the series’ radius of convergence.

∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ

Textbook Question

Finding a Sequence’s Formula

In Exercises 13–30, find a formula for the nth term of the sequence.

2, 6, 10, 14, 18, …Every other even positive integer

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Textbook Question

Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]

Textbook Question

Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ / (1 + √n)]

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