notes8 - The comparison test Motivation: We now know, for...

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The comparison test Motivation: We now know, for example, X k =1 1 k 3 converges by the p -test. What can we say about, for example X k =1 1 k 3 + 10 ? Since k 3 + 10 > k 3 for every k , we know 1 k 3 + 10 < 1 k 3 for every k . Math 9C Summer 2011 (UCR) Pro-Notes October 10, 2011 1 / 1
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So this says 1 1 3 + 10 + 1 2 3 + 10 + ··· + 1 n 3 + 10 < 1 1 3 + 1 2 3 + ··· + 1 n 3 . Another way to write this is n X k =1 1 k 3 + 10 < n X k =1 1 k 3 And since this hold for every single n , we can take the limit as n goes to and get X k =1 1 k 3 + 10 = lim n →∞ n X k =1 1 k 3 + 10 < lim n →∞ n X k =1 1 k 3 = X k =1 1 k 3 . But since the series on the right is finite, this means the series on the left is finite which is the same thing as saying n X k =1 1 k 3 + 10 converges. Math 9C Summer 2011 (UCR) Pro-Notes October 10, 2011 2 / 1
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X k =3 1 k diverges. It’s the harmonic series ( p = 1). If we wanted to know about, say X k =3 1 k - 2 we do the same thing, but in reverse. Since 1 k < 1 k - 2 for all k we know n X k =3 1 k < n X k =3 1 k - 2 And since the left side goes to when n goes to , the right side also goes to as n goes to . In other words
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notes8 - The comparison test Motivation: We now know, for...

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