2. Advice I’ve given students: If the series does not look like anything but you’re being asked to evaluate it, try partial fractions and see if you get something telescoping.

3. The limit comparison test asks whether the terms of two series are “proportional in the limit.” The ratio and root tests ask, in two ways, whether the terms of one series are “geometric in the limit.” The geometric series with terms gives ratio r between successive terms (), leading to the ratio test, and the nth root of its nth term is r times the nth root of c (assuming c is positive, and otherwise taking the negation of the series, which has the same convergence behavior), which limits to r, leading to the root test. This is why the cutoff point for convergence and divergence is 1 – that is what it is for geometric series. The distinction, that at limit 1 we don’t know the behavior, is because this is something only “geometric in the limit.”

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• Note that the sequence has limit 0 but the function oscillates. This is why you can use convergence of the function to conclude convergence of the sequence of points along its graph, but not vice-versa. Intuitively, the function must “connect the dots” for its behavior to match the sequence’s (but the sequence can never be wilder than the function).
• Let a_n = n and b_n = -n. These sequences each diverge (to ) but their sum is constantly 0. That is why the Limit Laws only allow you to transfer convergence of individual sequences to their sums, products, etc., and not the reverse.
• Consider the series with terms 1, -1/2, 2/3, -1/3, 1/2, -1/4, 2/5, -1/5, … It is an alternating series whose terms limit to zero. If we sum consecutive pairs of terms, however, we obtain 1/2, 1/3, 1/4, 1/5, …: the harmonic series, which we know diverges. This is why the Alternating Series Test requires the magnitude of the terms decrease to zero.

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