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In this lesson, we will learn how to test the convergence of many series using the root test.
Q1:
A series ∞ 𝑎 satisfies l i m → ∞ | 𝑎 | = 1 .
What can we conclude about the convergence of the series?
Q2:
Consider the series ∞ 𝑛 = 0 𝑛 𝑎 , where 𝑎 = ( 𝑛 + 1 ) 6 𝑛 𝑛 2 𝑛 .
Calculate l i m 𝑛 → ∞ 𝑛 | 𝑎 | . 1 𝑛
Hence, determine whether the series converges or diverges.
Q3:
Consider the series ∞ 𝑛 = 3 𝑛 2 − 𝑛 3 𝑛 + 1 .
Is this an alternating series?
Is this series absolutely convergent, conditionally convergent, or divergent?
Q4:
Consider the series , where .
Calculate .
Q5:
Consider the series 1 + 1 2 + 1 2 + 1 2 + ⋯ √ 2 √ 3 √ 4 , where the term 𝑎 = 1 2 𝑛 √ 𝑛 .
What is l i m 𝑛 → ∞ 𝑛 + 1 𝑛 | 𝑎 | | 𝑎 | ?
What is l i m 𝑛 → ∞ √ 𝑛 𝑛 1 2 ?
Use L’Hopital’s rule to determine the value of the limit l i m l n 𝑛 → ∞ 𝐴 𝑥 √ 𝑥 where 𝐴 > 0 is a constant.
What does the previous result tell you about the values of √ 𝑛 and 𝑛 l o g 2 2 where 𝑛 ≥ 1 is an integer?
Is this series convergent or divergent?
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