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In this lesson, we will learn how to identify whether a series is divergent or convergent using the comparison test.
Q1:
Use the comparison test to decide whether the series β π = 1 π ο 2 3 7 π + 1 . 1 is convergent or divergent.
Q2:
Let π = π + 1 π β π + 4 π + 2 8 π 2 3 2 . As π β β , we see that π β 1 π π , which suggests that β π = 1 π ο π is a divergent series. Verify this by finding the first integer π where π > 1 π π . You should check that this inequality remains true for all larger π .
Q3:
Consider the series β π = 1 2 ο π π = 1 1 + 2 4 + 3 9 + β― s i n s i n s i n s i n .
Do the terms π π of this sequence tend to 0 as π β β ?
Is the series convergent or divergent?
Q4:
Suppose β π = 1 π ο π is a series with the property that there is an integer π such that π π > 1 0 π β 8 for all indices π β₯ π .
Does it follow that all the terms of the series are positive?
Does it follow that the number of negative terms is finite?
Is the series convergent?
Is the series convergent if the inequality is π π < 1 0 2 π β 8 instead?
Q5:
Use the comparison test to determine whether is convergent or divergent.
Q6:
Use the comparison test to decide whether the series β π = 1 π β π ο οΌ 1 + 1 π ο π is convergent or divergent.
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