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Unformatted text preview: for . Hence, for , (Method 2: Ordinary Comparison Test) Let , Then for . Since converges, it follows that converges. Another way: for . Since converges, it follows that converges. (Method 3: Ratio Test) . It follows from Ratio Test that the series converges. (Method 4: Root Test) Noting that , we have . It follows that converges. ( b ) Since and diverges, it follows from Limit Comparison Test that diverges. ( c ) Let . Then is continuous since is continuous on and for . Also, , so f is decreasing. Moreover, Hence, and converges. It follows from Integral Test that converges. ( d ) Let . Then . Hence, . By Ratio Test, we have ( i ) If x <1, converges. ( ii ) If x >1, diverges. When x =1, the series becomes . It is known that when p >1, converges and when , diverges. • About this document . .. Next: About this document Kunquan Lan Wed Feb 9 09:35:54 EST 2000...
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This note was uploaded on 12/16/2009 for the course MATH 1014 taught by Professor Ganong during the Fall '09 term at York University.
 Fall '09
 ganong
 Math

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