Answered: 3 (n + 1)2, 8시

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.CR: Chapter Review

Problem 35E: Find three positive geometric means between 2 and 8

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Question

find a formula for the nth partial sum of the series
and use it to determine whether the series converges or diverges. If a
series converges, find its sum.

Expert Solution

Step 1

The formula for the nth partial sum is already given in the question itself:

3/n² - 3/(n+1)²

Step 2

Let's unfold the series:

n = 1; a₁ = 3/1² - 3/2²

n = 2; a₂ = 3/2² - 3/3²

n = 3; a₂ = 3/3² - 3/4²

and so on.

When we add these terms sequentially to get:

Series, S = a₁ + a₂ + a₃ + ....= 3/1² - 3/2²+ 3/2² - 3/3² + 3/3² - 3/4² + .....

Second term cancels the third one, fourth term cancels the fifth one and so on. Only the first term survives.

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