Intereting Posts

Evaluate the integral $\int^{\frac{\pi}{2}}_0 \frac{\sin^3x}{\sin^3x+\cos^3x}dx$
Volume of neighborhood of the curve
Does the little-oh relation remain if $f(x)$ and $g(x)$ both integrate or differentiate?
Counting invariant subspaces of a Vector space
How can I prove the maximum number of edges?
$\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample
Borel-Cantelli Lemma “Corollary” in Royden and Fitzpatrick
Compact operators on an infinite dimensional Banach space cannot be surjective
How to solve the general sextic equation with Kampé de Fériet functions?
When do we have $Rad(I)=I$ for an ideal $I$ of a ring $R$?
Definite Integral of $e^{ax+bx^c}$
The set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal
Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)
Apparently cannot be solved using logarithms
Ternary Quadratic Forms

I’ve been struggling with figuring out how to add powers of $i$.

For example, the result of $i^3 + i^4 + i^5$ is $1$. But how do I get the result of $i^3 + i^4 + … + i^{50}$? Writing it all down would be pretty mundane…

It has to do something with division by 4, since the “power cycle” of $i$ repeats every fourth power.

- Why is the MacLaurin series proof for eulers formula $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $ valid?
- $1/i=i$. I must be wrong but why?
- Complex number calculation
- Solving $(z+1)^5 = z^5$
- Where is the fallacy? $i=1$?
- “Where” exactly are complex numbers used “in the real world”?

Thank you for any clues.

- Finding the sum of the infinite series whose general term is not easy to visualize: $\frac16+\frac5{6\cdot12}+\frac{5\cdot8}{6\cdot12\cdot18}+\cdots$
- Closed form for $\sum_{n=1}^\infty \left(e-\left(1+\frac{1}{n}\right)^n \right)^2$?
- Using the Banach Fixed Point Theorem to prove convergence of a sequence
- Easy proof for sum of squares $\approx n^3/3$
- A question concerning dot product of sequences with a specific asymptotic growth.
- EGF of rooted minimal directed acylic graph
- How to prove $\sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5-\cdots}}}}}}} = \frac{2+\sqrt 5 +\sqrt{15-6\sqrt 5}}{2}$
- About Banach Spaces And Absolute Convergence Of Series
- Sequence is periodic $x_{n+2}=|x_{n+1}-x_{n-1}|$
- What was the motivation for the complex plane?

From $i^2=-1$ you get $i^{4n}=1$, $\quad i^{4n+1}=i$, $\quad i^{4n+2}=-1$ and $i^{4n+3}=-i$.

Then you just count your positive and negative multiples of $1$ and $i$.

In particular, $i^3+i^4+\cdots+i^{50}=0$.

Observing that $i^{3}+i^{4}+\ldots +i^{50}$ is a *geometric progression* with ratio $i$, first term $i^3$ and $50-3+1=48$ terms, we have

$i^{3}+i^{4}+\ldots +i^{50}=i^{3}\times \dfrac{1-i^{50-3+1}}{1-i}=i^{3}\times

\dfrac{1-i^{48}}{1-i}=i^{2}i\times \dfrac{1-(i^{2})^{24}}{1-i}$

$=-i\dfrac{1-(-1)^{24}}{1-i}=-i\dfrac{1-1}{1-i}=0$

Edit: “arithmetic” corrected to “geometric”

**HINT** $\rm\quad\quad i^3 + \: i^4 \; + \:\;\cdots\;\: + \; i^k = 0\ \:\iff\: k\:\equiv\: 2 \:\pmod 4$

Generally, suppose that $\rm\: z \:$ has order $\rm m>1\:.$ Therefore $\rm\; z^n = 1 \iff\ m\:|n\;\;\:$ hence:

**LEMMA** $\quad\rm z^j + z^{j+1} + \:\cdots + z^k = 0\;\; \iff \rm\: k \:\equiv\;\; j \:-\: 1 \:\pmod m $

**Proof**: $\;\;\;\;\rm \displaystyle z^j \ (1+z+\cdots + z^{k-j}) \;=\; z^j \: \frac{1-z^{k-j+1}}{1-z} = 0 \;\iff\; \rm m\:|\:k-j+1\quad\;$

We have $$i^{3} + i^{4} + i^{5} = 1 = i^{3} + i^{4} + i^{5} + i^{6} + i^{7} + i^{8} + i^{9} = i^{3} + i^{4} + \cdots +i^{4n+1}$$

Now $i^{50}=1 \times -1$, therefore we have the sum is $0$.

- Why 4 is not a primitive root modulo p for any prime p?
- How to find out whether linear programming problem is infeasible using simplex algorithm
- Proving if $F^{-1} $ is function $\Rightarrow F^{-1}$ is $1-1$?
- (Certain) colimit and product in category of topological spaces
- A $2 \times 2$ matrix $A$ such that $A^n$ is the identity matrix
- Numbers of the form $\frac{xyz}{x+y+z}$, second question
- Constructing a NFA for the following language
- Why is it not known if Mill's constant is rational or irrational?
- Fourier transform of $\frac{1}{f(t)}$
- How to prove that there exists $g(x)$ such $\int_{0}^{1}g(x)dx\ge\frac{1}{2}\int_{0}^{1}f(x)dx$
- Cardinality of the set of ultrafilters on an infinite Boolean algebra
- conditional expectation of brownian motion
- dice problem – throws necessary for sum multiple of n
- Why random variable is defined as a mapping from the sample space?
- If two Riemannian manifolds can be isometrically immersed in each other, are they isometric?