Department

MathematicsCourse Code

MAT136H1Professor

allStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**1 pages of the document.**11.6 Infinite Sequences & Series

Absolute Convergence & Ratio and Root Tests

Question #2 (Medium): Convergence by Ratio Test

Strategy

By the word “ratio” test, it takes and the next term expression and takes a ratio of the two so

that:

1)

means

is absolutely convergent (ie. convergent)

2)

, or means

is divergent

3)

means no conclusion can be drawn from the Ratio Test about convergence or

divergence of

All series expression falls into one of the three categories, thus depending on the outcome of the ratio

test, its convergence, or divergence, or no conclusion can be drawn based on the test.

Sample Question

Determine if the series is absolutely convergent conditionally convergent, or divergent.

Solution

To put into ratio test, is just taking the series expression inside the summation as it is, thus

The ratio text requires the next term expression , thus it is then

simply replacing

by wherever occurs.

Now that these two expressions are written down, the ratio can be taken:

then it can be simplified so that limit value

becomes more obvious:

Since this is less than 1, the series is absolutely convergent, thus the series

is convergent.

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