Q. 18 The Root Test: Let 鈭慿=1鈭瀉k聽... [FREE SOLUTION]

Chapter 7: Q. 18 (page 656)

The Root Test: Let ${鈭�}_{k = 1}^{鈭�} a_{k}$ be a series with nonnegative terms, and let $p = \lim_{k 鈫� 鈭�}_$
If 蟻 < 1, the series _.
If 蟻 > 1, the series _.
If 蟻 = 1, _.

Short Answer

Expert verified

Let ${鈭�}_{k = 1}^{鈭�} a_{k}$ be a series with nonnegative terms, and let localid="1649361486379" $p = \lim_{k 鈫� 鈭�} \sqrt[k]{a_{k}}$ .

If $p < 1$ , the series diverges.

If $p > 1$ , the series converges.

If $p = 1$ , the test is inconclusive.

Step by step solution

Step 1. Given Information

The given test is the root test.

Step 2. Explanation

To perform the root test, find $p = \lim_{k 鈫� 鈭�} \sqrt[k]{a_{k}}$
If $p < 1$ , the series is convergent.
If $p > 1$ , the series is divergent.
But if the limit is 1, the test cannot guarantee any behavior.

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The Root Test: Let ${鈭�}_{k = 1}^{鈭�} a_{k}$ be a series with nonnegative terms, and let $p = \lim_{k 鈫� 鈭�}_$
If 蟻 < 1, the series _.
If 蟻 > 1, the series _.
If 蟻 = 1, _.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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91影视

The Root Test: Let 鈭�k=1鈭�akbe a series with nonnegative terms, and letp=limk鈫�鈭�___If 蟻 < 1, the series ___.If 蟻 > 1, the series ___.If 蟻 = 1, ___.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Probability and Statistics

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

The Root Test: Let ${鈭�}_{k = 1}^{鈭�} a_{k}$ be a series with nonnegative terms, and let $p = \lim_{k 鈫� 鈭�}_$
If 蟻 < 1, the series _.
If 蟻 > 1, the series _.
If 蟻 = 1, _.