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Jan 10th, 2015

As n increases...

If we look on the top line of the fraction,

n is slowly increasing

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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