This question in itself isn't the tough part; its the series we're given that's really strange.
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Investigate the convergence (absolute and conditional) for the series u_n given by
1/(ln2) - 1/(2ln3) - 1/(2ln4) + 1/(2²ln5) + 1/(2²ln6) + 1/(2²ln7) + 1/(2²ln8) - 1/(2³ln9) - ...
- 1/(2³ln16) + 1/(2ln17) + ...
where the terms are positive, negative, positive, ... in blocks of 1, 2, 4, 8, ..., 2^m, ... so
u_n=(-1/2)^m [1/ln(n+1)] for 2^mn<2^(m+1), m=0,1,2,...
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Please show your work. (Note that those ellipses are intentional)
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Investigate the convergence (absolute and conditional) for the series u_n given by
1/(ln2) - 1/(2ln3) - 1/(2ln4) + 1/(2²ln5) + 1/(2²ln6) + 1/(2²ln7) + 1/(2²ln8) - 1/(2³ln9) - ...
- 1/(2³ln16) + 1/(2ln17) + ...
where the terms are positive, negative, positive, ... in blocks of 1, 2, 4, 8, ..., 2^m, ... so
u_n=(-1/2)^m [1/ln(n+1)] for 2^mn<2^(m+1), m=0,1,2,...
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Please show your work. (Note that those ellipses are intentional)