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We can see the first part (-1)^k does nothing than to alternate the sign of the outcome. The next part (1/2k) will progressively get smaller as k increases. This will be then subtracted from the previous positive number making that number smaller.
I figure T will be between 1/4 and 1/2 since the smallest base term is 1/4 (1/2-1/4) and adding 10 smaller terms shouldn't make it exceed 1/2