Chapter 7: Q. 17 (page 656)

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

Short Answer

Expert verified

Let ${鈭�}_{k = 1}^{鈭�} a_{k}$ be a series with nonnegative terms, and let localid="1649361444772" $p = \lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}}$ .

If $p < 1$ , the series converges.

If $p > 1$ , the series diverges.

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

Step by step solution

Step 1. Given Information

The given test is the ratio test.

Step 2. Explanation

The ratio test is used in the case of geometric series or series with terms like $a^{n}, n! e t c$ .
To perform the ratio test, find
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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91影视

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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91影视

The Ratio Test: Let 鈭�k=1鈭�akbe a series with nonnegative terms, and let p=limk鈫�鈭�___.If p<1, the series ___.If p>1, the series ___.If p=1, ___.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Geometry

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

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