Evaluate each series or state that it diverges. ∑k=2^∞((3)/(8))^3 k

Name: Problem 76, Chapter 10: Calculus: Early Transcendentals
Uploaded: 2020-05-29T10:37:47-08:00
Duration: 2 min 3 s
Channel: Vipender Yadav
Description: Evaluate each series or state that it diverges. ∑k=2^∞((3)/(8))^3 k

step 1 Here in this problem we have to evaluate the series which is Sigma k equals to 2 to infinite 3 d

Question

Evaluate each series or state that it diverges. $$\sum_{k=2}^{\infty}\left(\frac{3}{8}\right)^{3 k}$$

   Evaluate each series or state that it diverges.
$$\sum_{k=2}^{\infty}\left(\frac{3}{8}\right)^{3 k}$$

Calculus: Early Transcendentals

William Briggs, Lyle… 3rd Edition

Chapter 10, Problem 76

Instant Answer

Solved by Expert Vipender Yadav

05/29/2020

Step 1

The series is given as $$\sum_{k=2}^{\infty}\left(\frac{3}{8}\right)^{3 k}$$ This is a geometric series where the first term (a) is $\left(\frac{3}{8}\right)^{3 \cdot 2} = \left(\frac{3}{8}\right)^6$ and the common ratio (r) is $\left(\frac{3}{8}\right)^3$. Show more…

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Evaluate each series or state that it diverges. $$\sum_{k=2}^{\infty}\left(\frac{3}{8}\right)^{3 k}$$

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Key Concepts

Geometric Series

A geometric series is an infinite series where each term is obtained by multiplying the previous term by a constant known as the common ratio. The general form is ? ar^k, and the series converges if the absolute value of the common ratio is less than 1.

Convergence Criteria for Infinite Series

Determining whether an infinite series converges involves checking specific conditions. For a geometric series, the series converges when the absolute value of the common ratio is less than 1. This concept is critical for deciding if an infinite sum has a finite value.

Index Shifting in Series

Index shifting is a technique used to adjust the starting point of a series to match standard forms. This adjustment facilitates the application of known formulas, like the sum of a geometric series, by appropriately identifying the initial term and the common ratio.

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