Skip to main content
Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.3.11
Chapter 10, Problem 10.3.11

9–15. Geometric sums Evaluate each geometric sum.


{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ

Verified step by step guidance
1
Identify the type of series given. The sum is a geometric series because each term is obtained by multiplying the previous term by a constant ratio.
Write down the general form of a geometric series sum: \(S_n = a \frac{1 - r^{n+1}}{1 - r}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms minus one.
Determine the first term \(a\) by substituting \(k=0\) into the term expression: \(a = \left(\frac{2}{5}\right)^{2 \cdot 0} = \left(\frac{2}{5}\right)^0\).
Identify the common ratio \(r\) by examining the factor that each term is multiplied by to get the next term. Since the exponent increases by 2 each time, \(r = \left(\frac{2}{5}\right)^2\).
Substitute \(a\), \(r\), and \(n=20\) into the geometric sum formula and simplify the expression to find the sum without calculating the final numerical value.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. The series has the form a + ar + ar² + ... + arⁿ, where a is the first term and r is the common ratio.
Recommended video:
06:00
Geometric Series

Sum Formula for a Finite Geometric Series

The sum of the first n+1 terms of a geometric series is given by S = a(1 - r^(n+1)) / (1 - r), provided r ≠ 1. This formula allows quick calculation of the sum without adding each term individually.
Recommended video:
06:00
Geometric Series

Evaluating Powers and Substitution

To evaluate the geometric sum, you must correctly compute powers of the common ratio and substitute values into the sum formula. Understanding exponentiation and careful substitution ensures accurate calculation of the sum.
Recommended video:
04:27
Substitution With an Extra Variable
Related Practice
Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) (k² / 4ᵏ)

66
views
Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k^(–1/5)

66
views
Textbook Question

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.


aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)

70
views
Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.


∑ (k = 1 to ∞) (1 + 2 / k)ᵏ

31
views
Textbook Question

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁. 

aₙ = 1 + sin(πn / 2)

63
views
Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) (((1/6)ᵏ + (1/3)ᵏ) × k⁻¹)

79
views