2. a. Simplify left(fracsqrt22 - fracsqrt22i ight)^20 b....

2. a. Simplify $left(frac{sqrt{2}}{2} - frac{sqrt{2}}{2}i ight)^{20}$ b. Test for convergence or divergence of the following sequences i. $u_n = left{frac{3n + 5^n}{n + 6^n} ight}$ ii. $u_n = left{frac{e^n}{11n^4} ight}$ c. Find the values of $a$ and $b$ in the equation below $frac{(1 + i)b - 2i}{i + 3} + frac{(2 - 3i)a + i}{3 - i} = i$

Transcript

00:02 Okay, so for the first one, let's write this guy here in polar form.

00:07 Now we have that square root of 2 over 2 minus square root of 2 over 2 multiplied by i is equal to e to the negative by i over 4.

00:20 So in particular this guy here is e to the negative 5 pi over multiplied by i.

00:32 Which is e to the negative pi i which is the same thing as e to the pi i which is just negative one so we are done with the first one well for the second one here we just need to notice that five to the n over six to the n goes to zero because five over six is less than one so this guy converges to zero for this one, we just need to notice that the exponential function grows much faster than a power function.

01:15 So this guy is going to go to infinity.

01:19 Finally, for c, we just need to simplify this complex number here.

01:27 Well, first of all, let's remember that if z is a complex number, then one or one of is just the conjugate of z over the length of z squared so in particular we got that 1 over i plus 3 is negative i plus 3 over the length squared of this guy which is square root of 10 similarly we are going to do the same for 3 to the negative i so we are going to have that this first piece here is negative i plus 3 multiplied by 1 plus i multiplied by b negative i plus 3 by negative 2i so we are going to have negative 2 negative 6 i over square root of 10 and here we're going to have what we're going to have 1 over square root of 10 multiplied by okay 3 plus i multiplied by 2 minus 3 i a and we're going to have plus 3 i here okay now let's rewrite this guy here in a more convenient form so here we're going to have 1 over square root of 10 multiplied by what multiplied by this one multiplied by this one is negative i b minus b plus 3 b plus 3 i b minus 2 minus 6 i and here we are going to have plus 6 i plus 6 i plus 2 oh, and everything is going to be multiplied by a.

04:15 So maybe we can do like this.

04:20 And finally, the last piece is 3i minus 1.

04:25 So plus 3i minus 1...

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