in will therefore also be slowly increasing in magnitude
and n to the i th power with also be increasing in magnitude
This is happening at a fairly constant rate, since if we factorise out n in the top bracket we get:
n( 1 + i + n^(i - 1) )
The interesting part of the sequence is the denominator.
i to the n means the square root of -1 to the nth power
If we have i squared then that is -1
But i to the power of 4 is 1
This idea repeats: i^6 is -1, but i^8 is 1
i.e. if n is a multiple of 4 then the end result is positive and therefore the result in the sequence is positive.
But if the power- if n- is a multiple of 2 but not of four then the result will be negative
Therefore the sequence will fluctuate up and down increasing in magnitude as the sequence continues- as the numerator is larger and larger and the constant flipping of the bottom turns it from positive to negative and back again.
The sequence is divergent.
Jan 10th, 2015
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