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Comparison Test
COMPARISON TEST Let c be a positive constant. Suppose and are positive term series and a_{n} ≤ cb_{n} for all n. (i) If converges then converges. (ii) If diverges then diverges. PROOF (i) Suppose converges to S. The Constant Rule gives
Each finite partial sum of is less than cS, Therefore, an infinite partial sum is less than cS and hence finite. It follows that converges. (ii) If diverges then cannot converge by part (i). To use the Comparison Test we compare a series whose convergence or divergence is unknown with one which is known.
The harmonic series diverges, whence the given series diverges. Sometimes the following comparison test is easier to use.


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