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Comparison Test
Assume that we have two types of series ∑an and ∑bn with an, bn ≥ 0 for all n and an ≤ bn for all n.
Then,
A. If ∑bn is convergent then this is ∑an.
B. If ∑an is divergent then this is ∑bn.
Alternatively, we have two series of positive terms and the terms of one of the series are all time larger than the terms of the other series. After that if the larger series is convergent the smaller series must as well be convergent. Similarly, if the smaller series is divergent as compared to the larger series must as well be divergent.
Note: That in order to apply this test we require both series to start at similar place.
(a) Determine the matrix that first rotates a two-dimensional vector 180° anticlockwise, and then per- forms a horizontal compression of the resulting vector by a factor 1/2 (leavi
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