# Further Maths - Complex Sums

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/complex-sums/ · Updated: 2022-03-30 · Tags: school, futher-maths, core-year-2, a-level

## See Also
- [Further Maths - Complex Numbers](https://ollybritton.com/notes/a-level/further-maths/topics/complex-numbers/)
- [Further Maths - Exponential Form of Complex Numbers](https://ollybritton.com/notes/a-level/further-maths/topics/exponential-form-of-complex-numbers/)
- [Further Maths - Trig Equations with Complex Numbers](https://ollybritton.com/notes/a-level/further-maths/topics/trig-equations-with-complex-numbers/)
- [Further Maths - Roots of Complex Numbers](https://ollybritton.com/notes/a-level/further-maths/topics/roots-of-complex-numbers/)

## Flashcards

### Closing the geometric sum
##### $$w + wz + wz^2 + \cdots + wz^{n-1}$$ What is this equal to??
$$
\frac{w(z^n - 1)}{z - 1}
$$
^geometric-sum-w-up-to-n-minus-one

##### $$\sum^{n - 1}_{r = 0} wz^r$$ What is this equal to??
$$
\frac{w(z^n - 1)}{z - 1}
$$
^geometric-sum-sigma-up-to-n-minus-one

##### $$w + wz + wz^2 + \cdots + wz^{n}$$ What is this equal to??
Remember the plus one! Don't you dare skip this card if you didn't say that explicitly!

$$
\frac{w(z^{n+1} - 1)}{z - 1}
$$
^geometric-sum-w-up-to-n

##### $$\sum^{n}_{r = 0} wz^r$$ What is this equal to??
Remember the plus one! Don't you dare skip this card if you didn't say that explicitly!

$$
\frac{w(z^{n+1} - 1)}{z - 1}
$$
^geometric-sum-sigma-up-to-n

### Strategy for complex geometric sums
##### What is the point of the complex sums of series topic??
Turning complicated series into simple expressions of $\sin$ and $\cos$.
^complex-sums-purpose

##### What are the two important results when working out complex geometric sums??
$$
z + \frac{1}{z} = 2\cos\theta
$$
$$
z - \frac{1}{z} = 2i\sin\theta
$$
^z-plus-minus-reciprocal-identities

##### $$\frac{-2}{e^{\frac{\pi i}{n}} - 1}$$ What might your next step be here??
You're trying to create the difference or sum of two opposite pairs, multiply the bottom by $e^{-\frac{\pi i}{2n}}$

$$
-2\frac{e^{-\frac{\pi i}{2n}}}{e^{\frac{\pi i}{2n}} - e^{-\frac{\pi i}{2n}}}
$$
^complex-sum-multiply-by-half-angle-exponential

##### What are the two techniques for making $z + 1/z$ or $z - 1/z$ terms appear from nothing in the complex sums topic??
* Factoring out
* Multiplying top and bottom
^techniques-creating-z-plus-reciprocal-terms

##### $$1 + z + z^2 + \cdots + z^{2n-1}$$ What is this equal to??
$$
\frac{1(z^{2n} - 1)}{z - 1}
$$
^geometric-sum-unit-up-to-2n-minus-one

##### $$1 + z + z^2 + \cdots + z^n$$ What is this equal to??
$$
\frac{1(z^{n+1} - 1)}{z - 1}
$$
^geometric-sum-unit-up-to-n

### Finishing a worked simplification
##### $$\frac{-2e^{-\frac{\pi}{2n}i}}{2i\sin\left( \frac{\pi}{2n} \right)}$$ How could you simplify this??
Use the fact that $-2/2i = i$

$$
\frac{ie^{-\frac{\pi}{2n}}}{\sin\left( \frac{\pi}{2n} \right)}
$$
^complex-sum-simplify-minus-two-over-two-i

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