Wesyu Posted September 29, 2004 Share Posted September 29, 2004 Anyone knows how to do these? http://www.math.mcgill.ca/jakobson/courses/ma262/a1.pdf Link to comment Share on other sites More sharing options...
bloodhound Posted September 29, 2004 Share Posted September 29, 2004 havent done this for a while . but here goes for the first one. use the ratio test see if the limit as n tends to infinity of [math]|a_{n+1}/a_{n}|[/math] exists. where in this case. [math]a_{n}=x^{n}\sum_{i=1}^{n}\frac{1}{i}[/math] so we have [math]|a_{n+1}/a_{n}|=|x\frac{\sum_{i=1}^{n+1}1/i}{\sum_{i=1}^{n}1/i}|[/math] which just converges to |x| as n tends to infinity therefore the series converges if |x|<1 i.e -1<x<1. when x = -1 . the series doesnt converge as [math]\sum_{i=1}^{n}1/i[/math] doesnt converge to 0. and if x =1 the seires doesnt converge as it is the standard harmonic series. Link to comment Share on other sites More sharing options...
bloodhound Posted September 29, 2004 Share Posted September 29, 2004 have to think a lot harder for the second one. its quite easy to show that it does indeed converge for some x . but what i dont know. it does defintely converge if -1/10 < x < 1/10 . but that is the radius of convergence for the comaprision series i used to check if the original series converges. Link to comment Share on other sites More sharing options...
bloodhound Posted September 29, 2004 Share Posted September 29, 2004 for the comparision for the second question , i used . [math]\sum_{n=1}^{\infty}\frac{10^{n}x^{n}}{n}[/math] Link to comment Share on other sites More sharing options...
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